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.
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