  We derive the Operator Product Expansion whose vacuum expectation value gives
the time-moments of the pseudoscalar heavy-light current-current correlator up
to and including terms in $\alpha_s^2$ multiplying
$\langle\overline{\psi}\psi\rangle/M^3$ and terms in $\alpha_s$ multiplying
$\langle \alpha_s G^2 \rangle/M^4$, where $M$ is the heavy-quark mass. Using
lattice QCD results for heavy-strange correlators obtained for a variety of
heavy quark masses on gluon field configurations including $u$, $d$ and $s$
quarks in the sea at three values of the lattice spacing, we are able to show
that the contribution of the strange-quark condensate to the time-moments is
very substantial. We use our lattice QCD time-moments and the OPE to determine
a value for the condensate, fitting the 4th, 6th, 8th and 10th time-moments
simultaneously. Our result, $\langle \overline{s}s
\rangle^{\overline{\text{MS}}}(2 \text{GeV}) = -(296(11) \,\mathrm{MeV})^3$,
agrees well with HPQCD's earlier, more direct, lattice QCD
determination~\cite{McNeile:2012xh}. As well as confirming that the $s$ quark
condensate is close in value to the light quark condensate, this demonstrates
clearly the consistency of the Operator Product Expansion for fully
nonperturbative calculations of matrix elements of short-distance operators in
lattice QCD.
