We present extensive experimental data on the gate-charge-periodic current {ital I}({ital Q}{sub {ital o}}) through a single-electron transistor as a function of the applied magnetic field. This device consists of a mesoscopic superconducting island which is coupled to two macroscopic superconducting leads through small tunnel junctions and to a capacitive gate. The behavior of the system exhibits two separate transitions as the magnetic field is increased. In the first, the {ital I}{minus}{ital Q}{sub {ital o}} curves cross from 2{ital e} to {ital e} periodicity in {ital Q}{sub {ital o}} while two-electron tunneling is still the dominant charge transport mechanism. In a second transition that occurs at higher magnetic fields, single-electron tunneling becomes the dominant mechanism. This transition from two-electron to single-electron tunneling shifts the maxima of the {ital I}-{ital Q}{sub {ital o}} curves by {ital e}/2 in {ital Q}{sub {ital o}}, and results in a significant increase in the current magnitude. Both transitions can be understood by considering how superconductivity in the leads and the island is affected by an increasing magnetic field. {copyright} {ital 1996 The American Physical Society.}