  In this paper we significantly improve estimates of distance to the X-ray
binary systems Cyg X-3 and GRS 1915+105. We report a highly accurate
trigonometric parallax measurement for Cyg X-3 using the VLBA at 43 GHz,
placing the source at a distance of 9.67+0.53-0.48 kpc. We also use Galactic
proper motions and line-of-sight radial velocity measurements to determine
3-dimensional (3D) kinematic distances to both systems, under the assumption
that they have low peculiar velocities. This yields distances of 8.95+-0.96 kpc
for Cyg X-3 and 9.4+-0.6 (statistical)+-0.8 (systematic) for GRS 1915+105. The
good agreement between parallax and 3D kinematic distances validates the
assumption of low peculiar velocities, and hence small natal kicks, for both of
the systems. For a source with a low peculiar velocity, given its parallax
distance, Cyg X-3 should have a Vlsr near -64+-5 km/s. Our measurements imply a
slightly higher inclination angle, and hence lower black hole mass for GRS
1915+105 than found from previous work by Reid et al (2014) and strengthen
arguments from X-ray polarization that Cyg X-3 would be an ultraluminous X-ray
source if viewed face-on.
