Answer:
Barack Obama was born on Friday, August 4, 1961, at 7:24 pm in Kapi'olani Maternity and Gynecological Hospital, in Honolulu, Oahu, Hawaii, USA.
Note:
Some earlier reports indicated the hospital was named the Queen's Medical Center. The name Kapi'olani is the same as one of the queens and the women's hospital was named for her. So the hospital has been called both names informally, but the proper name and the name on the official birth certificate is Kapiolani.