  We present a method to measure the critical temperature of the island of a
superconducting single electron transistor. The method is based on a sharp
change in the slope of the zero-bias conductance as a function of temperature.
We have used this method to determine the superconducting phase transition
temperature of the Nb island of an superconducting single electron transistor
with Al leads. We obtain $T_\mathrm{c}^\mathrm{Nb}$ as high as 8.5 K and gap
energies up to $\Delta_\mathrm{Nb}\simeq 1.45$ meV. By looking at the zero bias
conductance as a function of magnetic field instead of temperature, also the
critical field of the island can be determined. Using the orthodox theory, we
have performed extensive numerical simulations of charge transport properties
in the SET at temperatures comparable to the gap, which match very well the
data, therefore providing a solid theoretical basis for our method.
