  Cold atom magnetometers exploit a dense ensemble of quanta with long
coherence times to realise leading sensitivity on the micrometer scale.
Configured as a Ramsey interferometer, a cold atom sensor can approach atom
shot-noise limited precision but suffers from fringe ambiguity, producing gross
errors when the field falls outside a narrow predefined range. We describe how
Hilbert-demodulated optical magnetometry can be realised on cold atom sensors
to provide field measurements both precise and unambiguous. Continuous
reconstruction of the Larmor phase allows us to determine the dc magnetic field
unambiguously in an unshielded environment, as well as measure ac variation of
the field, in a single shot. The ac measurement allows us to characterize, and
then neutralise, line-synchronous magnetic interference, extending
reconstruction times. Using $1.6 \times 10^6$ $^{87}$Rb atoms in a volume of
$(68 \,\mathrm{\mu m})^3$, we measure a test field to be $ 86.0121261(4) \;
\mathrm{\mu T}$ in a single shot, achieving dc sensitivity of 380 fT in a
duration of 1000 ms. Our results demonstrate that Hilbert-demodulated optical
readout yields metrologically-significant sensitivity without the fringe
ambiguity inherent to Ramsey interferometry.
