  SrTiO$_{3}$, a quantum paraelectric, becomes a metal with a superconducting
instability after removal of an extremely small number of oxygen atoms. It
turns into a ferroelectric upon substitution of a tiny fraction of strontium
atoms with calcium. The two orders may be accidental neighbors or intimately
connected, as in the picture of quantum critical ferroelectricity. Here, we
show that in Sr$_{1-x}$Ca$_{x}$TiO$_{3-\delta}$ ($0.002<x<0.009$,
$\delta<0.001$) the ferroelectric order coexists with dilute metallicity and
its superconducting instability in a finite window of doping. At a critical
carrier density, which scales with the Ca content, a quantum phase transition
destroys the ferroelectric order. We detect an upturn in the normal-state
scattering and a significant modification of the superconducting dome in the
vicinity of this quantum phase transition. The enhancement of the
superconducting transition temperature with calcium substitution documents the
role played by ferroelectric vicinity in the precocious emergence of
superconductivity in this system, restricting possible theoretical scenarios
for pairing.
