We have investigated the intrinsic Josephson properties in slightly underdoped (Hg,Re)Ba 2 Ca 2 Cu 3 O y [Hg(Re)1223] intrinsic Josephson junctions (IJJs) with dimension of 1.0 x 1.5 x 0.11 µm 3 . The current-voltage characteristics of the IJJs exhibit clear multiple branches with subgap structures similar to those of other cuprate superconductors. The switching current distributions, P(I), from the zero-voltage to a nonzero-voltage state in the current-biased IJJs agree well with the theoretical curves of the thermally assisted escape model at temperatures above ≈ 5 K. The plasma frequency, f p , of the IJJs is estimated to be 1.3 THz from the fluctuation-free critical current density of 2.0 x 10 5 A/cm 2 , which is one of the highest among cuprate superconductors, reflecting the high T c and a relatively low anisotropy of the Re doped Hg system. The P(I) gradually becomes independent of temperature below ≈ 5 K, which suggests a crossover of the escape process from thermal activation to quantum tunneling at such a high temperature.