Jessica, it is possible to convert from B-1 to H-1B. Normally a lawyer will submit the H-1B petition on your behalf, and the most important thing is that you are legal in the country the whole time. IE, don't overstay your visa and then apply for H-1B.
The purpose of a visa is to enter the U.S. You will not get a visa in your passport in the United States. Once your
H-1B petition is approved, you are legally in the United States and can work. If you ever leave the U.S., you must go home and get a visa at the U.S. consulate or embassy in order to return. This should be a straight forward process.
K.Man.