We demonstrate ground-state cooling of a trapped ion using radio-frequency (rf) radiation. This is a powerful tool for the implementation of quantum operations, where rf or microwave radiation instead of lasers is used for motional quantum state engineering. We measure a mean phonon number of $\overline{n}=0.13$ after sideband cooling, corresponding to a ground-state occupation probability of 88%. After preparing in the vibrational ground state, we demonstrate motional state engineering by driving Rabi oscillations between the $|n=0⟩$ and $|n=1⟩$ Fock states. We also use the ability to ground-state cool to accurately measure the motional heating rate and report a reduction by almost 2 orders of magnitude compared with our previously measured result, which we attribute to carefully eliminating sources of electrical noise in the system.