The discovery of interstellar NH is reported in the diffuse clouds toward Zeta Per and HD 27778 from high-resolution high S/N spectra of the NH A 3Pi-X 3Sigma (0,0) absorption band near 3358 A. These observations represent the first detection of this molecule anywhere in the interstellar medium. Measured equivalent widths of 0.37 + or - 0.08 and 1.1 + or - 0.3 mA for the NH R1(0) lines toward Zeta Per and HD 27778 correspond to respective NH column densities of 9.0 x 10 to the 11th and 2.7 x 10 to the 12th/sq cm. N(NH) in the Zeta Per cloud is close to previous upper limits and much greater than that predicted by current models of the gas-phase nitrogen chemistry. The incorporation of a more rapid rate for the reaction N(+) + H2 yields NH(+) + H in these models may provide an effective gas-phase route to explain the NH abundance. A contribution involving the direct or indirect production of NH from grain surface reactions is also possible. In any case, these NH measurements have additional implications for a better understanding of the observed CN abundances in diffuse interstellar clouds.