What agency maintains marriage records in DR for American citizens wedded there? What is the easiest (i.e. most efficient) way to get copies of these records?
The same one that maintains them for Dominicans that wed there just like your state agency would for Dominicans that wed where you are. Other than in person good luck getting through those bureauracies from home.
Searching for a particular marriage record can be very time consuming unless one knows the precise date of the marriage and the precise location of the Oficialia (registry) where the marriage was recorded.
