  In recent years, a handful of ``dark" binaries have been discovered with a
non-luminous compact object. Astrometry and radial velocity measurements of the
bright companion allow us to measure the post-supernova orbital elements of
such a binary. In this paper, we develop a statistical formalism to use such
measurements to infer the pre-supernova orbital elements, and the natal kick
imparted by the supernova (SN). We apply this formalism to the recent discovery
of an X-ray quiet binary with a black hole, VFTS 243, in the Large Magellanic
Cloud. Assuming an isotropic, Maxwellian distribution on natal kicks and using
broad agnostic mass priors, we find that kick velocity can be constrained to
$V_k < 72$ km/s and the dispersion of the kick distribution to $\sigma_k < 68 $
km/s at 90 % confidence. We find that a Blaauw kick cannot be ruled out and
}that at least about $0.6 M_{\odot}$ was lost during the supernova with 90 %
confidence. The pre-SN orbital separation is found to be robustly constrained
to be around $0.3$ AU.
