Using the inverse scattering method we construct an exact stationary asymptotically flat 4+1-dimensional vacuum solution describing Black Saturn: a spherical black hole surrounded by a black ring. Angular momentum keeps the configuration in equilibrium. Black saturn reveals a number of interesting gravitational phenomena: (1) The balanced solution exhibits 2-fold continuous non-uniqueness for fixed mass and angular momentum; (2) Remarkably, the 4+1d Schwarzschild black hole is not unique, since the black ring and black hole of black saturn can counter-rotate to give zero total angular momentum at infinity, while maintaining balance; (3) The system cleanly demonstrates rotational frame-dragging when a black hole with vanishing Komar angular momentum is rotating as the black ring drags the surrounding spacetime. Possible generalizations include multiple rings of saturn as well as doubly spinning black saturn configurations.

48 pages, 35 figures. v2: clarifications + new discussion of two sectors of balanced solutions + corrections.