  SrTiO$_3$ exhibits a superconducting dome upon doping with Nb, with a maximum
critical temperature \mbox{$T_\mathrm{c} \approx 0.4$~K}. Using microwave
stripline resonators at frequencies from 2 to 23~GHz and temperatures down to
0.02~K, we probe the low-energy optical response of superconducting SrTiO$_3$
with charge carrier concentration from 0.3 to $2.2\times 10^{20}$~cm$^{-3}$,
covering the majority of the superconducting dome. We find single-gap
electrodynamics even though several electronic bands are superconducting. This
is explained by a single energy gap $2\Delta$ due to gap homogenization over
the Fermi surface consistent with the low level of defect scattering in
Nb-doped SrTiO$_3$. Furthermore, we determine $T_\mathrm{c}$, $2\Delta$, and
the superfluid density as a function of charge carrier concentration, and all
three quantities exhibit the characteristic dome shape.
