Where is an amniocentesis usually done? At a hospital?

Usually office. Amniocentesis is the procedure by which fluid is removed from the amniotic sac. It is done to study chromosomes, perform genetic tests, check lung maturity, etc. In general, it is performed by an obgyn specialist or a maternal-fetal medicine subspecialist in the office or clinic. It might also be performed on labor and delivery or the antepartum ward. Ask the OB provider what is best in your case.
Doctor's office. In amniocentesis, a small amount of amniotic fluid is withdrawn from the womb. This is typically done to perform genetic testing on the baby, rarely to reduce the fluid volume in extreme oligohydramnios. It is generally performed in an outpatient setting such as a doctor's office or outpatient clinic.