  Zero-field nuclear magnetic resonance (NMR) provides complementary analysis
modalities to those of high-field NMR and allows for ultra-high-resolution
spectroscopy and measurement of untruncated spin-spin interactions. Unlike for
the high-field case, however, universal quantum control -- the ability to
perform arbitrary unitary operations -- has not been experimentally
demonstrated in zero-field NMR. This is because the Larmor frequency for all
spins is identically zero at zero field, making it challenging to individually
address different spin species. We realize a composite-pulse technique for
arbitrary independent rotations of $^1$H and $^{13}$C spins in a two-spin
system. Quantum-information-inspired randomized benchmarking and state
tomography are used to evaluate the quality of the control. We experimentally
demonstrate single-spin control for $^{13}$C with an average gate fidelity of
$0.9960(2)$ and two-spin control via a controlled-not (CNOT) gate with an
estimated fidelity of $0.99$. The combination of arbitrary single-spin gates
and a CNOT gate is sufficient for universal quantum control of the nuclear spin
system. The realization of complete spin control in zero-field NMR is an
essential step towards applications to quantum simulation,
entangled-state-assisted quantum metrology, and zero-field NMR spectroscopy.
