  Doped strontium titanate SrTiO$_3$ (STO) is one of the most dilute
superconductors known today. The fact that superconductivity occurs at very low
carrier concentrations is one of the two reasons that the pairing mechanism is
not yet understood, the other is the role played by the proximity to a
ferroelectric instability. In undoped STO, ferroelectric order can in fact be
stabilized by substituting $^{16}$O with its heavier isotope $^{18}$O. Here we
explore the superconducting properties of doped and isotope-substituted
SrTi$(^{18}$O$_{y}^{16}$O$_{1-y})_{3-\delta}$ for $0\le y \le 0.81$ and carrier
concentrations between $6\times 10^{17}$ and $2\times 10^{20}$ cm$^{-3}$
($\delta<0.02$). We show that the superconducting $T_c$ increases when the
$^{18}$O concentration is increased. For carrier concentrations around $5\times
10^{19}$~cm$^{-3}$ this $T_c$ increase amounts to almost a factor $3$, with
$T_c$ as high as 580~mK for $y=0.74$. When approaching SrTi$^{18}$O$_3$ the
maximum $T_c$ occurs at a much smaller carrier densities than for pure
SrTi$^{16}$O$_3$. Our observations agree qualitatively with a scenario where
superconducting pairing is mediated by fluctuations of the ferroelectric soft
mode.
