  The alloys of non-centrosymmetric superconductor, Re$_3$W, which were
reported to have an $\alpha$-Mn structure [P. Greenfield and P. A. Beck, J.
Metals, N. Y. \textbf{8}, 265 (1959)] with $T_\mathrm{c}=9 $K were prepared by
arc melting. The ac susceptibility and low-temperature specific heat were
measured on these alloys. It is found that there are two superconducting phases
coexisting in the samples with $T_\mathrm{c1}\sim9 $K and $T_\mathrm{c2}\sim7
$K, both of which have a non-centrosymmetric structure as reported previously.
By analyzing the specific heat data measured in various magnetic fields, we
found that the absence of the inversion symmetry does not lead to the deviation
from a s-wave pairing symmetry in Re$_3$W.
