The negatively-charged silicon-vacancy (SiV$^-$) color center in diamond has
recently emerged as a promising system for quantum photonics. Its
symmetry-protected optical transitions enable creation of indistinguishable
emitter arrays and deterministic coupling to nanophotonic devices. Despite
this, the longest coherence time associated with its electronic spin achieved
to date ($\sim 250$ ns) has been limited by coupling to acoustic phonons. We
demonstrate coherent control and suppression of phonon-induced dephasing of the
SiV$^-$ electronic spin coherence by five orders of magnitude by operating at
temperatures below 500 mK. By aligning the magnetic field along the SiV$^-$
symmetry axis, we demonstrate spin-conserving optical transitions and
single-shot readout of the SiV$^-$ spin with 89% fidelity. Coherent control of
the SiV$^-$ spin with microwave fields is used to demonstrate a spin coherence
time $T_2$ of 13 ms and a spin relaxation time $T_1$ exceeding 1 s at 100 mK.
These results establish the SiV$^-$ as a promising solid-state candidate for
the realization of scalable quantum networks.