  The origin of superconductivity in bulk SrTiO$_{3}$ is a mystery, since the
non-monotonous variation of the critical transition with carrier concentration
defies the expectations of the crudest version of the BCS theory. Here,
employing the Nernst effect, an extremely sensitive probe of tiny bulk Fermi
surfaces, we show that down to concentrations as low as $5.5 \times
10^{17}cm^{-3}$, the system has both a sharp Fermi surface and a
superconducting ground state. The most dilute superconductor currently known
has therefore a metallic normal state with a Fermi energy as little as 1.1 meV
on top of a band gap as large as 3 eV. Occurrence of a superconducting
instability in an extremely small, single-component and barely anisotropic
Fermi surface implies strong constraints for the identification of the pairing
mechanism.
