  We have obtained high resolution Ca II and Na I absorption line spectra
toward 68 OB stars in the direction of the Vela Supernova Remnant. The stars
lie at distances of 190 -- 2800 pc as determined by Hipparcos and spectroscopic
parallax estimations. The presence of high velocity absorption attributable to
the remnant along some of the sight lines constrains the remnant distance to
250+/-30 pc. This distance is consistent with several recent investigations
that suggest that the canonical remnant distance of 500 pc is too large.
