Large-order perturbation theory has been applied, for the first time, to the Stark effect for ${\mathrm{H}}_{2}^{+}$, yielding the Rayleigh-Schr\"odinger ground-state eigenvalue (polarizability) series through twentieth order; previous expansions were limited to fourth order. The calculations were performed nonadiabatically (i.e., without invoking the Born-Oppenheimer approximation) by means of the perturbational-variational Rayleigh-Ritz formalism. The leading terms of the Rayleigh-Schr\"odinger polarizability series so obtained provide the most accurate values thus far determined for ${\ensuremath{\alpha}}_{\mathrm{zz}}$ and ${\ensuremath{\gamma}}_{\mathrm{zzzz}}$.