Is there passport control between Belfast and Dublin?
As there is no physical border between the Republic of Ireland and Northern Ireland you will not pass through UK immigration control if arriving to Belfast through Dublin. As such, your passport will not be endorsed with a UK entry stamp.
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Visit the Embassy of Ireland's website for the most current visa information. You must have a valid passport to enter Ireland. U.S. citizens can enter visa-free for tourism or business stays of up to 90 days.
In most cases, you'll go through customs and immigration after your flight first arrives in a new country, but there are exceptions. For example, some countries have mutual agreements intended to help speed travelers through the process, so you may go through customs and immigration before you board.