The high-latitude Galactic H I cloud toward the extragalactic radio source 3C 225 is characterized by very narrow 21 cm emission and absorption indicative of a very low H I spin temperature of about 20 K. Through high-resolution optical spectroscopy, we report the detection of strong, very narrow Na I absorption corresponding to this cloud toward a number of nearby stars. Assuming that the turbulent H I and Na I motions are similar, we derive a cloud temperature of 20 (+6, -8) K (in complete agreement with the 21 cm results) and a line-of-sight turbulent velocity of 0.37+/-0.08 km/s from a comparison of the H I and Na I absorption linewidths. We also place a firm upper limit of 45 pc on the distance of the cloud, which situates it well inside the Local Bubble in this direction and makes it the nearest-known cold diffuse cloud discovered to date.

11 pages, 3 figures, accepted for publication in ApJ Letters