Shipping Intelligence. Arrivals, Moreton Bay Courier, Brisbane, 11 May 1861, p 2, http://nla.gov.au/nla.news-article3718487:

May 10.—Yarra, Yarra, steamer, 560 tons, Capt. Bell from Sydney. Passengers—Right Rev. Dr. Quinn, Rev. Mr. Casse [Cusse], Rev. Mr. Moloney, [...]

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Local Intelligence, The North Australian, Ipswich and General Advertiser, Ipswich, 11 June 1861, p 3, http://nla.gov.au/nla.news-article77434183:

The Rev James Molony, Minister of the Roman Catholic Church, Brisbane, is registered as a Minister of Religion for the celebration of Marriages.

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Marriages, Queensland Guardian, Brisbane, 17 July 1861, p 3, http://nla.gov.au/nla.news-article290951220:

Cumming—Maguire—On the 5th July, at North Brisbane, by the Rev. James Moloney, George Cumming, of Drayton, to Margaret Maguire, of Brisbane.

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Fr T.J. Linane, From Abel to Zundolovich, Vol. 1, 1979, p 52: "Molony, James. Irish. Victoria to 1861."

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Denis M. Long, Pioneer and Early Priests, Archdiocese of Brisbane, Deceased or retired by 1929, 2021, p 91:
Priests of whom we know little

The names of these priests appear on the Sick and Infirm Priests’ Sustentation Society prayer card for deceased priests, but of whom our knowledge is limited. [...]

James Moloney was in Ipswich in 1863. A priest of that name also served in Melbourne. No more records of this priest are available.

Reproduced by permission of Fr Denis Long.