Shipping Intelligence. Hobson's Bay. Arrivals—Oct. 16, The Argus, Melbourne, 18 October 1875, p 4, http://nla.gov.au/nla.news-article7422584:

Pera, R.M.S.S., 2,119 tons, Robert Methven, commander, from Point de Galle 24th ult [...] Passengers [...] For Sydney—saloon: [...] Rev. Dr. Quin, Rev. A. Petre, Rev. W. Rice [...]

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Bathurst, Freeman's Journal, Sydney, 20 November 1875, p 12, http://nla.gov.au/nla.news-article115474319:

On Sunday, the 14th instant, (reports the Independent) the Feast of St. Lawrence O'Toole, was celebrated the tenth anniversary of the consecration of the Bishop of Bathurst. [...] Pontifical High Mass was celebrated by his Lordship, assisted by the Very Rev. Dr. Byrne as deacon, and the Rev. Mr. O'Riordan as sub-deacon. The Rev. W. Rice, Cloyne, Ireland, acted as assistant priest, and the Very Rev. Dr. McAuliffe, President of St. Stanislaus' College, as Master of Ceremonies. [...] After the Gospel, the Rev. W. Rice ascended the pulpit and delivered a very beautiful and eloquent discourse.

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Shipping. Clearance.—April 7, Sydney Morning Herald, 9 April 1877, p 4, http://nla.gov.au/nla.news-article13394803:

Zealandia (s.), 3000 tons, Captain Chevalier, for San Francisco. Passengers—[...] Rev. W. Rice, Rev. J Nelan, [...]

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Ireland, Freeman's Journal, Sydney, 20 October 1877, p 6, http://nla.gov.au/nla.news-article115375258:

Among the passengers by the White Star steamer Britannic, which sailed from New York on July 21, and arrived off Cloyne on the 29th ult., was the Rev. William Rice, of the Diocese of Cloyne, after an absence of two years in Australia, where he was collecting funds for the new Cathedral in Queenstown.

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Irish death record, https://www.irishgenealogy.ie/files/civil/deaths_returns/deaths_1930/04921/4335988.pdf: Superintendent Registrar's District Mallow, Registrar's District Mitchelstown No. 1. 1930. Deaths Registered in the District of Mitchelstown No. 1 in the Union of Fermoy in the County of Cork. No. 174: On 5 February 1930 at Mitchelstown, death of Archdeacon William Rice, male, bachelor, 94 years, Parish Priest; of Myocardial Degeneration, 2 years to my knowledge, Cardiac Dilatation, 8 days, certified; informant: James Rice, present at death, of Ballindangan, 5 March 1930.

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Loss to Cloyne Diocese. Death of Ven. Archdeacon Rice, Irish Independent, Dublin, 6 February 1930, p 11:

The death of the venerable Archdeacon Rice, P.P., V.G., has evoked feelings of deepest regret amongst the parishioners of Mitchelstown and throughout the diocese of Cloyne. Born in 1837 at Kilworth, he was at an early age sent to Mount Melleray, where he studied, housed in huts with the other students. From Melleray he passed to Fermoy and from thence to Maynooth, where he had a brilliant career. He was ordained in 1862, and his first mission was to Glanworth, whence he went to Ballyvourney, and late in 1864 to Ahabulloge, and in 1869 to Fermoy.

From 1875 to 1877 he was in Australia and New Zealand collecting funds for St. Colman's Cathedral, Cobh. Whilst in Australia he was appointed Administrator of Fermoy, and in 1883 he was appointed P.P. of Ballymacoda. In 1898 he was transferred to Mitchelstown. He was appointed Archdeacon in 1917, and Vicar-General in 1923.

He was a life-long student and kept up his studies until his last illness.

The late Archdeacon was a brother of Rev. T. Rice, P.P., Ballyhea, who died in 1889, and of Richard Rice solicitor and Coroner, who held amongst other inquests that on the victims of the Mitchelstown shooting, and who died in 1925.

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Day of Mourning in Mitchelstown. Pastor's Funeral. Bishop's Tribute to Late Archdeacon Rice, Evening Echo, Cork, 7 February 1930, p 1:

There were remarkable scenes of respect and mourning at Mitchelstown this morning when the obsequies of the late Ven. Archdeacon Rice took place.

Business was completely suspended for the day. The funeral ceremonies were timed for 11 a.m., but as early as eight o'clock people were making their way to the parish church, where the remains of the revered pastor had been placed on a catafalque before the High Altar the previous evening. There were crowded congregations at all the Masses. After nine o'clock, at which hour the majority of the business houses customarily open, the town presented an appearance of deep mourning, all establishments being closed and shuttered, and the only animation being the arrival of motors. These brought hundreds of people, priests and laity, from the towns and villages between Cork on one side, and Limerick on the other. Many others came by traps and bicycles, and despite the bitterly cold weather, there were hundreds of people in the church for a considerable time before the obsequies began.

Most Rev. Dr. Browne, the venerable Bishop of Cloyne, who had travelled from Cove the previous day, and stayed at St. Colman's College, Fermoy for the night, arrived at Mitchelstown early and the solemn Office commenced punctually at 11 o'clock. The scene in the interior was a memorable one. Over sixty priests, representing every part of the Diocese of Cloyne and many parts of Cork Diocese, were in the sanctuary, and there was an enormous congregation, composed of people of all ages and classes, many of whom had travelled long distances. The elaborate scaffolding in the interior, erected in connection with the repairs to the roof, bore silent testimony to the late Archdeacon's practical interest in Church affairs to the end of his life.

His Lordship was attended at the throne [...]

At the conclusion of the Mass, his Lordship addressed the congregation from the throne, and expressed sincere sympathy with the people of Mitchelstown and the people of the diocese generally, on the great loss they had sustained by the Archdeacon's death. His Lordship referred to the Archdeacon's great influence, not only by his preaching, but by his wonderful example. As an example of the Archdeacon's humility, Most Rev. Dr. Brown mentioned that he would not permit any celebration of his Golden Jubilee. He also alluded to the Archdeacon's work in looking after the repairs of the parish church.

The funeral procession through the town was an impressive spectacle. The coffin was borne down the aisle and into the grounds nephews of the Archdeacon—Messrs. Coroner E.A. Rice, solr., Fermoy; W. St. Clair Rice, solr., Midleton; T.A. Rice, Fermoy, and J. Rice, do. The cortege was led by Rev. J. Rice, C.C., who acted as cross-bearer. A large number of pupils of the Presentation Convent followed; then came a corps of the Christian Brothers' pupils, a large section of Children of Mary; next the clergy in soutanes and surplices, many of them in canonicals; then the pall-bearers carrying the coffin, which, in accordance with the Archdeacon's wishes, was of a very simple design, and, finally, a great crowd of townspeople and sympathies generally.

While the priests chanted prayers for the dead, the procession passed slowly down Chapel Hill, and thence to the Square, through which it wended solemnly, many other members of the public looking on with heads bared. The funeral returned to the church by two routes—Chapel Hill and Convent Hill—and here the remains were received by Rev. Fr. Scannell. His Lordship the Bishop read the last prayers at the graveside in the burial ground attached to the church, the melancholy scene being reverently watched by a great concourse.

The funeral arrangements were excellently carried out by Mr. Denis Forrest, Royal Hotel, Mitchelstown.

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Gossip, Freeman's Journal, Sydney, 10 April 1930, p 19, http://nla.gov.au/nla.news-article121657808:

The death has occurred of the Venerable Archdeacon William Rice, Vicar-General of Cloyne, Ireland, and parish priest of Mitchelstown, at the age of 92. He was born at Kilworth in 1837 and was educated at Mount Melleray, continuing his studies at Maynooth. After his ordination in 1862, Father Rice was appointed to Glanworth. Here he remained only a short period and then went to Ballyvourney, Aghabulloge and Fermoy. In 1875 he was chosen by the Right Rev. Dr. McCarthy, Bishop of Cloyne, to collect funds for Cove (Queenstown) Cathedral, in Australia and New Zealand. As a result of his labors the sum of £4000 was collected, and he returned to Ireland in 1877. In 1883 Father Rice was appointed parish priest of Ballymacoda, where he was made a Canon of the Cathedral Chapter. On being transferred to Mitchelstown 15 years later, he was appointed Vicar Forane. He was made Archdeacon in 1917 and Vicar General in 1923.

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T.W. Campbell, The Priests of the Diocese of Armidale: A Necrology, Braddon ACT, 2005:
Rice, William

Listed in E J Doody, Acta et Statuta 1951, Appendix VI, Obituary List, in the section headed 'supplementary list of priests who have also worked in the diocese', but with no details other than the period he reputedly worked in the diocese, in this case 1876.
A William Rice travelled around colonial Australia in that period seeking funds for the construction of a cathedral in the Irish city of Queenstown. There is no indication that he worked as a priest of the diocese. It is possible that he travelled and preached within the area of the diocese during his quest for funds.
(Sources: Doody, MDHC A-Z R26).

(Reproduced by permission of Tom Campbell.)