  To any smooth compact manifold $M$ endowed with a contact structure $H$ and
partially integrable almost CR structure $J$, we prove the existence and
uniqueness, modulo high-order error terms and diffeomorphism action, of an
approximately Einstein ACH (asymptotically complex hyperbolic) metric $g$ on
$M\times (-1,0)$.
  We consider the asymptotic expansion, in powers of a special defining
function, of the volume of $M\times (-1,0)$ with respect to $g$ and prove that
the log term coefficient is independent of $J$ (and any choice of contact form
$\theta$), i.e., is an invariant of the contact structure $H$.
  The approximately Einstein ACH metric $g$ is a generalisation of, and
exhibits similar asymptotic boundary behaviour to, Fefferman's approximately
Einstein complete K\"ahler metric $g_+$ on strictly pseudoconvex domains. The
present work demonstrates that the CR-invariant log term coefficient in the
asymptotic volume expansion of $g_+$ is in fact a contact invariant. We discuss
some implications this may have for CR $Q$-curvature.
  The formal power series method of finding $g$ is obstructed at finite order.
We show that part of this obstruction is given as a one-form on $H^*$. This is
a new result peculiar to the partially integrable setting.
