That's a very good question. World Net Daily says that Kapi*olani Health Foundation in Hawaii has published a letter from our President wherein the President claims to have been born at their hospital. Follow the links, then read the letter, posted below.
Which is fine, on its face. However, there are conflicting reports, including this aritle from The Rainbow Edition (.pdf file), a laboratory school in Hawaii. It reports that our President was born in Queens Hospital in Hawaii.
So, the question become, where was the President born? I understand that even Snopes has weighed in on the matter. They believe that he was born in Hawaii. Fair enough.
Still, questions remain and there's one good way to put these questions to rest. Release the original birth certificate. It it turns out that our President was not born in the United States, a whole bag of constitutional issues will be raised and President Obama's name will surely be written in national history.
Curmudgeonly and Skeptical is doing yeoman's work on this question.