New high-resolution infrared and UV/visible spectra of (107)AgH, (109)AgH, (107)AgD, and (109)AgD have been recorded with a Fourier transform spectrometer. The new line positions are combined with published microwave and older electronic A (1)Sigma(+)-X (1)Sigma(+) data and used, first in a decoupled analysis of the X state alone, and then in a global multi-isotopologue analysis which yields comprehensive descriptions of both the X (1)Sigma(+) and A (1)Sigma(+) states of all four isotopologues of AgH. While the A state was long believed to be heavily perturbed, it is shown that its irregular spectrum merely reflects an unusual potential function shape. A direct fit of all data to appropriate radial Hamiltonians yields analytic potential-energy functions and Born-Oppenheimer breakdown radial functions for the ground X (1)Sigma(+) and A (1)Sigma(+) states.