Google translate gives me: Duōmǐníjiā gònghéguó.
But I wondered if the chinese use a smaller word or lop-off the Republic at the end?
First of all, let's confirm the translation. If you look for "Dominican Republic" on Wikipedia, and then choose the chinese page, the result is:
Duō mǐ ní jiā gòng hé guó
So, yes, it's the right translation. If you don't like Wikipedia as the only source, this is confirmed by the YouDao and CEDICT dictionaries.
I tried searching for "多米尼加" only (in both of those named above) and the result is simply "Dominica". Note that "Dominican Republic" and "Dominica" refer to two different countries, so since I think the same ambiguity is present in Chinese, you should keep the whole expression in order to be understood unambiguously.
多米尼加 is an acceptable short form of 多米尼加共和国.
Though the Wikipedia page lists 多米尼加共和国 as the primary name for the Dominican Republic, the sidebar says, "通称：多米尼加", i.e., "Commonly called 多米尼加." The Baidu page on 多米尼加 is about the Dominican Republic, and Google results for 多米尼加 return pages about the Dominican Republic. A quick search reveals that 多米尼加 is also common in news reports about the country. (E.g., see here.)
(Dominica, the other country Alenanno mentioned, is transliterated as 多米尼克, as Wikipedia attests. It is true that it used to be called 多米尼加联邦, but 多米尼克 is now preferred.*)
*EDIT: This claim is from the Wikipedia entry I linked to above. The Chinese original is as follows:
As the last sentence says, the usage of 多米尼克 is now completely standard in both China and Taiwan.