We have developed an <em>in-situ</em> index of refraction profile n(z) for cold polar ice, using the transit times of radio signals broadcast from an englacial transmitter to 1&ndash;5 km distant radio-frequency receivers, deployed at depths up to 200 m. For propagation through a non-uniform medium, Maxwell&rsquo;s equations generally admit two ray propagation solutions from a given transmitter, corresponding to a direct path (D) and a refracted or reflected path (R); the measured D vs. R timing differences (dt(D,R)) are determined by the refractive index profile. We constrain n(z) near the South Pole, where the Askaryan Radio Array (ARA) neutrino observatory is located, by simulating D and R ray paths via ray tracing and comparing simulations to measured dt(D,R) values. We demonstrate that our dt(D,R) timing data strongly favors a glaciologically-motivated three-phase densification model rather than the single exponential scale height models typically employed by in-ice radio neutrino detectors. Effective volume simulations for a detector of ARA station antenna depths yield a 14 % increase in neutrino sensitivity over a neutrino energy range of 10¹⁸&minus; 10²¹ eV using the three-phase model compared to the single exponential.