  During its 2003 outburst, the black-hole X-ray transient H1743-322 produced
two-sided radio and X-ray jets. Applying a simple and symmetric kinematic model
to the trajectories of these jets, we determine the source distance, 8.5 \pm
0.8 kpc, and the inclination angle of the jets, 75 \pm 3 degrees. Using these
values, we estimate the spin of the black hole by fitting its RXTE spectra,
obtained during the 2003 outburst, to a standard relativistic accretion-disk
model. For its spin, we find a*=0.2 \pm 0.3 (68% limits); -0.3 < a* < 0.7 at
90% confidence. We rule strongly against an extreme value of spin: a* < 0.92 at
99.7% confidence. H1743-322 is the third known microquasar (after A0620-00 and
XTE J1550-564) that displays large-scale ballistic jets and has a moderate
value of spin. Our result, which depends on an empirical distribution of black
hole masses, takes into account all known sources of measurement error.
