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 \pm 0.08 km s ^ { -1 } 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 .