The use of a generalized Lanczos technique for the accurate diagonalization of lattice-gauge-theory Hamiltonians is explored. The starting ansatz wave function is a Gaussian one which solves the theory exactly in the weak-coupling limit. Results are given for compact U(1) lattice gauge theory in 3+1 dimensions, on ${4}^{3}$ and ${6}^{3}$ spatial lattices, and for a range of couplings in the continuum region. With 8 Lanczos states the precision for ground-state energies obtained ranges (depending on lattice size and coupling) from 1 part in ${10}^{6}$ to 1 part in ${10}^{8}$. .AE