  We systematically investigate the normal and superconducting properties of
non-centrosymmetric Re$_{6}$Zr using magnetization, heat capacity, and
electrical resistivity measurements. Resistivity measurements indicate
Re$_{6}$Zr has poor metallic behavior and is dominated by disorder. Re$_6$Zr
undergoes a superconducting transition at $T_{\mathrm{c}} =
\left(6.75\pm0.05\right)$ K. Magnetization measurements give a lower critical
field, $\mu_{0}H_{\mathrm{c1}} = \left(10.3 \pm 0.1\right)$ mT. The
Werthamer-Helfand-Hohenberg model is used to approximate the upper critical
field $\mu_{0}H_{\mathrm{c2}} = \left(11.2 \pm 0.2\right)$ T which is close to
the Pauli limiting field of 12.35 T and which could indicate singlet-triplet
mixing. However, low-temperature specific-heat data suggest that Re$_{6}$Zr is
an isotropic, fully gapped s-wave superconductor with enhanced electron-phonon
coupling. Unusual flux pinning resulting in a peak effect is observed in the
magnetization data, indicating an unconventional vortex state.
