  We have performed the first direct measurement of two resonances of the
$^7$Be($\alpha,\gamma$)$^{11}$C reaction with unknown strengths using an
intense radioactive $^7$Be beam and the DRAGON recoil separator. We report on
the first measurement of the 1155 and 1110 keV resonance strengths of $1.73 \pm
0.25(stat.) \pm 0.40(syst.)$ eV and $125 ^{+27}_{-25}(stat.) \pm 15(syst.)$
meV, respectively. The present results have reduced the uncertainty in the
$^7$Be($\alpha,\gamma$)$^{11}$C reaction rate to $\sim$ 9.4-10.7% over T =
1.5-3 GK, which is relevant for nucleosynthesis in the neutrino-driven outflows
of core-collapse supernovae ($\nu p$-process). We find no effect of the new,
constrained reaction rate on $\nu p$-process nucleosynthesis.
