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 $2.2times 10^{20}$~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 $2Delta$ due to gap homogenization over the Fermi surface consistent with the low level of defect scattering in Nb-doped SrTiO$_3$. Furthermore, we determine $T_mathrm{c}$, $2Delta$, and the superfluid density as a function of charge carrier concentration, and all three quantities exhibit the characteristic dome shape.
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