PrV$_2$Al$_{20}$ is a rare example of a heavy fermion system based on strong hybridization between conduction electrons and nonmagnetic quadrupolar moments of the cubic $Gamma_3$ ground doublet. Here, we report that a high-quality single crystal of PrV$_2$Al$_{20}$ exhibits superconductivity at $T_{rm c}=$ 50 mK in the antiferroquadrupole-ordered state under ambient pressure. The heavy fermion character of the superconductivity is evident from the specific heat jump of $Delta C/T sim 0.3$ J/mol K$^2$ and the effective mass $m^*/m_0 sim 140$ estimated from the temperature dependence of the upper critical field. Furthermore, the high-quality single crystals exhibit double transitions at $T_{rm Q}$ = 0.75 K and $T^{*}$ = 0.65 K associated with quadrupole and octapole degrees of freedom of the $Gamma_3$ doublet. In the ordered state, the specific heat $C/T$ shows a $T^3$ dependence, indicating the gapless mode associated with the quadrupole and/or octapole order. The strong sensitivity to impurity of the superconductivity suggests unconventional character due to significant quadrupolar fluctuations.
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