A compact aneutronic fusion engine will enable more challenging exploration missions in the solar system. This engine uses a deuterium-helium-3 reaction to produce fusion energy by employing a novel field-reversed magnetic field configuration (FRC). The FRC has a simple linear solenoidal coil configuration yet generates higher plasma pressures for a given magnetic field than other designs. Waste heat generated from bremsstrahlung and synchrotron radiation is recycled to maintain the fusion temperature. The charged reaction products, augmented by additional propellant, are exhausted through a magnetic nozzle. As an example, we present a mission to deploy the James Webb Space Telescope from LEO to an L2 halo orbit using a one MW compact aneutronic fusion rocket engine. The engine produces 20 N of thrust with an exhaust velocity of 55 km/s and has a specific power of 0.77 kW/kg.