We report a comprehensive study of the noncentrosymmetric superconductor Mo$_3$P. Its bulk superconductivity, with $T_c = 5.5$ K, was characterized via electrical resistivity, magnetization, and heat-capacity measurements, while its microscopic electronic properties were investigated by means of muon-spin rotation/relaxation ($mu$SR) and nuclear magnetic resonance (NMR) techniques. In the normal state, NMR relaxation data indicate an almost ideal metallic behavior, confirmed by band-structure calculations, which suggest a relatively high electron density of states, dominated by the Mo $4d$-orbitals. The low-temperature superfluid density, determined via transverse-field $mu$SR and electronic specific heat, suggest a fully-gapped superconducting state in Mo$_3$P, with $Delta_0= 0.83$ meV, the same as the BCS gap value in the weak-coupling case, and a zero-temperature magnetic penetration depth $lambda_0 = 126$ nm. The absence of spontaneous magnetic fields below the onset of superconductivity, as determined from zero-field $mu$SR measurements, indicates a preserved time-reversal symmetry in the superconducting state of Mo$_3$P and, hence, spin-singlet pairing.
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