As far back as 1817 Father Flynn called at the settlement, on his voyage from England to Sydney, and administered the sacrament, and four years later Father Connolly arrived and commenced his work. He celebrated mass at Mr. Curr’s store in Argyle-street, and there the congregation consisted of nine persons. Services were subsequently held in the old building in Elizabeth-street, known to us in later years as “Rats’ Castle.” Father Connolly having obtained a grant of 14 acres of land at the angle of Harrington and Brisbane streets, called it Mount Carmel, and in February, 1822, a small wooden church was commenced. It was finished in the following year, and dedicated to God, under the invocation of St. Virgilius.
The Mercury, 31 December 1904

ROMAN CATHOLIC CHAPEL
THE Resident Roman Catholic Clergyman in Van Diemen’s Land, in Furtherance of the Object with which he is intrusted, under the Authority of His Majesty’s Government, being desirous to put in Progress the Building of a suitable Place of Worship for the Roman Catholics, with the Sanction of His Honor the Lieutenant Governor, respectfully calls upon such of the Inhabitants as are disposed to give their Aid to this desirable Object, as connected with the good Order of a large Portion of the Persons already in Van Diemen’s Land, and of others probably yet to arrive ; and he respectfully requests to add, that a Subscription List is open at the under-mentioned Places :
At the Office of Edward Lord, Esq. ; of Messrs. Kemp and Co. ; of Edward Curr, Esq. ; at the House of P. A. Mulgrave, Esq. and at the Gazette Office, Hobart Town.
At the House of T. A, Lascelles, Esq. New Norfolk ; James Gordon, Esq. Pitt Water; and G. W. Gunning, Esq. Coal River.
At Port Dalrymple:-Thomas Archer, Esq. J. P.; Mr. Commissary Roberts ; T, C, Simpson, Esq. ; and James Cox, Esq. J. P.
Where the Contributions of all Persons will be thankfully received.
It is considered scarcely necessary to state, that the Catholic Community, for whose particular Accommodation and Good this Undertaking is about to be commenced, will be expected to give their Aid in every Shape in which it can be useful, and suitable to their Condition.
Hobart Town Gazette, 13 April 1822

I’m not sure on actual location on the chapel, but it appears to have been close to the corner of Harrington & Brisbane Sts. In 1831, an advertiser gives his location as “situated in Brisbane-street, at the corner of Harrington street, (opposite the Roman Chatholic Chapel). In 1833, a cottage is advertised for sale at the “premises in Brisbane street facing the Catholic Chapel”. In 1834, another advertiser gives his location as “opposite the Catholic Chapel, Harrington Street”.