Abstract. 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 × (−1,0). We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × (−1,0) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact form θ), 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 Kähler 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. 1.