  The synthesis and detailed characterization of the noncentrosymmetric
superconductor Re$_{6}$Hf using powder x-ray diffraction (XRD), magnetization,
transport and thermodynamic measurements are reported. XRD confirms the crystal
structure of Re$_{6}$Hf to be cubic noncentrosymmetric $\alpha$-\textit{Mn}
with a lattice parameter a = 9.6850(3)\text{\AA}. Resistivity, DC and AC
magnetization confirms the bulk superconducting transition T$_{c}^{onset} \sim
$ ~ 5.96 K. The lower critical field H$_{C1}(0)\approx$ 5.6 mT and upper
critical H$_{C2}(0)$ was found to be around 12.2 T which is greater than the
Pauli limiting field, indicating unconventional behavior in Re$_{6}$Hf and
leads to the possibility of triplet pairing. A sharp discontinuity in the
specific heat data was observed around T$_{c}$ indicating bulk
superconductivity, and the normalized specific heat jump is $\Delta
C_{el}$/$\gamma_{n}$T$_{c}$ = 1.53 which is in close agreement with BCS value.
Superconducting electronic specific heat data is fitted reasonably well for an
exponential relation with Sommerfeld coefficient ($\gamma$) showing a linear
relation with magnetic field suggesting that in Re$_{6}$Hf dominant pairing
channel is s-wave with isotropic superconducting gap parameter without nodes.
