  The noncentrosymmetric superconductor Mo$_3$Rh$_2$N, with $T_c = 4.6$ K,
adopts a $\beta$-Mn-type structure (space group $P$4$_1$32), similar to that of
Mo$_3$Al$_2$C. Its bulk superconductivity was characterized by magnetization
and heat-capacity measurements, while its microscopic electronic properties
were investigated by means of muon-spin rotation and relaxation ($\mu$SR). The
low-temperature superfluid density, measured via transverse-field (TF)-$\mu$SR,
evidences a fully-gapped superconducting state with $\Delta_0 = 1.73
k_\mathrm{B}T_c$, very close to 1.76 $k_\mathrm{B}T_c$ - the BCS gap value for
the weak coupling case, and a magnetic penetration depth $\lambda_0 = 586$ nm.
The absence of spontaneous magnetic fields below the onset of
superconductivity, as determined by zero-field (ZF)-$\mu$SR measurements, hints
at a preserved time-reversal symmetry in the superconducting state. Both TF-and
ZF-$\mu$SR results evidence a spin-singlet pairing in Mo$_3$Rh$_2$N.
