  We present a novel, highly efficient yet accurate analytical approximation
for the Green's function of a Holstein polaron. It is obtained by summing all
the self-energy diagrams, but with each self-energy diagram averaged over the
momenta of its free propagators. The result becomes exact for both zero
bandwidth and for zero electron-phonon coupling, and is accurate everywhere in
the parameter space. The resulting Green's function satisfies exactly the first
six spectral weight sum rules. All higher sum rules are satisfied with great
accuracy, becoming asymptotically exact for coupling both much larger and much
smaller than the free particle bandwidth. Comparison with existing numerical
data also confirms this accuracy. We use this approximation to analyze in
detail the redistribution of the spectral weight as the coupling strength
varies.
