We have measured the nonlinear refractive-index coefficient gamma interferometrically at 355 nm in fused silica and in BK-10, a borosilicate crown glass. These measurements are the first reported direct ultraviolet measurements of the nonlinear index of refraction in any transparent glass. Our results are gamma = (2.5 +/- 1.2) x 10(-16) cm(2)/W and gamma = (1.7 +/- 0.8) x 10(-16) cm(2)/W, respectively, for fused silica and BK-10.