  Experimental evidence suggests that superconductivity in SrTiO$_3$ is
mediated by a soft transverse ferroelectric mode which, according to
conventional theories, has negligible coupling with electrons. A
phenomenological Rashba type coupling has been proposed on symmetry arguments
but a microscopic derivation is lacking. Here we fill this gap and obtain a
linear coupling directly from a minimal microscopic model of the electronic
structure. We find that the effective electron-electron pairing interaction has
a strong momentum dependence. This yields an unusual situation in which the
leading $s$-wave channel is followed by a sub-leading $p$-wave state which
shows a stronger pairing instability than the $d$-wave state. The bare Rashba
coupling constant is estimated for the lowest band of doped SrTiO$_3$ with the
aid of first-principles computations. Extrapolating the estimation to the
multi-band regime we show that it can explain bulk superconductivity. For
densities where the Meissner effect has not been observed an extra softening
due to inhomogeneities is needed to explain the zero resistance state.
