We systematically investigate the normal and superconducting properties of noncentrosymmetric Re₆Zr using magnetization, heat capacity, and electrical resistivity measurements. Resistivity measurements indicate Re₆Zr has poor metallic behavior and is dominated by disorder. Re₆Zr undergoes a superconducting transition at T_c = (6.75 ± 0.05) K. Magnetization measurements give a lower critical field, μ₀H_c1 = (10.3 ± 0.1) mT. The Werthamer-Helfand-Hohenberg model is used to approximate the upper critical field μ₀H_c2 = (11.2 ± 0.2)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 Re₆Zr 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.