  We report a comprehensive experimental investigation of the magnetic
structure of the cycloidal phase in Ca$_3$Ru$_2$O$_7$, which mediates the spin
reorientation transition, and establishes its magnetic phase diagram. In zero
applied field, single-crystal neutron diffraction data confirms the scenario
deduced from an earlier resonant x-ray scattering study: between $46.7$~K $< T
< 49.0$~K the magnetic moments form a cycloid in the $a-b$ plane with a
propagation wavevector of $(\delta,0,1)$ with $\delta \simeq 0.025$ and an
ordered moment of about 1 $\mu_{\rm{B}}$, with the eccentricity of the cycloid
evolving with temperature. In an applied magnetic field applied parallel to the
$b$-axis, the intensity of the $(\delta,0,1)$ satellite peaks decreases
continuously up to about $\mu_0 H \simeq 5$ T, above which field the system
becomes field polarised. Both the eccentricity of the cycloid and the
wavevector increase with field, the latter suggesting an enhancement of the
anti$-$symmetric Dzyaloshinskii$-$Moriya interaction via magnetostriction
effects. Transitions between the various low-temperature magnetic phases have
been carefully mapped out using magnetometry and resistivity. The resulting
phase diagram reveals that the cycloid phase exists in a temperature window
that expands rapidly with increasing field, before transitioning to a polarised
paramagnetic state at 5 T. High-field magnetoresistance measurements show that
below $T\simeq 70$ K the resistivity increases continuously with decreasing
temperature, indicating the inherent insulating nature at low temperatures of
our high-quality, untwinned, single-crystals. We discuss our results with
reference to previous reports of the magnetic phase diagram of
Ca$_3$Ru$_2$O$_7$ that utilised samples which were more metallic and/or
poly-domain.
