Hi
I have been there. Official records of Births & Marriages started in 1864. They would have been retained in Dublin at that time because there was no Republic of Ireland then. However, rcords of the Births & Marriages are also retained in Belfast now. They can search for anyone born in the 'Six Counties' from 1864 for these, and the records of Deaths began a little later. The Mormon Church genealogical facilities have copies of such records for the whole of Ireland on reels. I use that facility quite often. Was there a specific name you wished to research? I may be able to look through some reels for you next week.
Kind regards
Ed.