The Kondo effect has been observed in a single gate-tunable atom. The measurement device consists of a single As dopant incorporated in a silicon nanostructure. The atomic orbitals of the dopant are tunable by the gate electric field. When they are tuned such that the ground state of the atomic system becomes a (nearly) degenerate superposition of two of the silicon valleys, an exotic and hitherto unobserved valley Kondo effect appears. Together with the "regular" spin Kondo, the tunable valley Kondo effect allows for reversible electrical control over the symmetry of the Kondo ground state from an SU(2) to an SU(4) configuration.