We present SpaceHopper, a three-legged, small-scale robot designed for future mobile exploration of asteroids and moons. The robot weighs 5.2kg and has a body size of 245mm while using space-qualifiable components. Furthermore, SpaceHopper's design and controls make it well-adapted for investigating dynamic locomotion modes with extended flight-phases. Instead of gyroscopes or fly-wheels, the system uses its three legs to reorient the body during flight in preparation for landing. We control the leg motion for reorientation using Deep Reinforcement Learning policies. In a simulation of Ceres' gravity (0.029g), the robot can reliably jump to commanded positions up to 6m away. Our real-world experiments show that SpaceHopper can successfully reorient to a safe landing orientation within 9.7 degree inside a rotational gimbal and jump in a counterweight setup in Earth's gravity. Overall, we consider SpaceHopper an important step towards controlled jumping locomotion in low-gravity environments.