The noncentrosymmetric superconductor Re$_6$Zr has attracted much interest due to the observation of broken time-reversal symmetry in the superconducting state. Here we report an investigation of the superconducting gap structure of Re$_6$Zr single crystals by measuring the magnetic penetration depth shift $Deltalambda(T)$ and electronic specific heat $C_e(T)$. $Deltalambda(T)$ exhibits an exponential temperature dependence behavior for $T~ll~T_c$, which indicates a fully-open superconducting gap. Our analysis shows that a single gap $s$-wave model is sufficient to describe both the superfluid density $rho_s(T)$ and $C_e(T)$ results, with a fitted gap magnitude larger than the weak coupling BCS value, providing evidence for fully-gapped superconductivity in Re$_6$Zr with moderate coupling.
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