A baby has 300 bones.
According to Henry Gray's Anatomy of the Human Body, commonly known as Gray's Anatomy there are approximately 270 bones in the body of a newborn. One of the people who answered this question stated 360 bones but I don't have any idea where that number came from.
Of course, the bones of a newborn are almost as soft as simple cartilage but, at during a nine month gestation, this cartilage takes the shape of bones and are classified as such generally. As an infant grows some of these 270 bones fuse together and eventually form approximately 206 bones by adulthood.
As an aside, even a 1st year anatomy student should be able to tell you that the actual amount of bones present in the human body can vary from individual to individual due to a number of factors (malformation, additional fusing, additional cartilage formation and subsequent hardening, etc.). So, the above numbers really are approximates.