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We report on superconductivity in single crystals of SrTiO$_{3-\delta}$ withcarrier densities $\textit{n} < 1.4 \times10^{18}cm^{-3}$, where only a singleband is occupied. For all samples in this regime, the resistive transitionoccurs at $T_{c} \approx 65 \pm 25 \ mK$. We observe a zero resistance statefor $\textit{n}$ as low as $1.03 \times10^{17}cm^{-3}$, and a partial resistivetransition for $\textit{n} = 3.85 \times10^{16}cm^{-3}$. We observe lowcritical current densities, relatively high and isotropic upper criticalfields, and an absence of diamagnetic screening in these samples. Our findingssuggest an inhomogeneous superconducting state, embedded within a homogeneoushigh-mobility 3-dimensional electron gas. $T_{c}$ does not vary appreciablywhen $\textit{n}$ changes by more than an order of magnitude, inconsistent withconventional superconductivity.

Superconductivity in the dilute single band limit in reduced Strontium Titanate