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I have vacationed in the Dominican Republic, and the only document I had to have was my passport.
Yes. To enter the Dominican Republic you will need your passport. period.
Diving in the Dominican is good all year. December is in the midst of the coolest weather with sun/cloud mix but temps average 78F.
Dominicans, must obtain a Visa to travel outside of the Dominican Republic. Since the Dominican Republic is it's own sovereign nation.
You need a passport to travel there from the US.