  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
$\Delta\lambda(T)$ and electronic specific heat $C_e(T)$. $\Delta\lambda(T)$
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.
