Weyl semimetals are a novel class of topological materials with unique electronic structures and distinct properties. HfRhGe stands out as a noncentrosymmetric Weyl semimetal with unconventional superconducting characteristics. Using muon-spin rotation and relaxation (µSR) spectroscopy and thermodynamic measurements, a fully gapped superconducting state is identified in HfRhGe that breaks time-reversal symmetry at the superconducting transition. This breaking can trigger a topological phase transition, as time-reversal symmetry protects the normal-state Weyl topology characterized by comprehensive first-principles calculations. Ginzburg-Landau analysis suggests an unconventional loop supercurrent superconducting state realized in HfRhGe. The presence of multiple Weyl points near the Fermi level and surface Fermi arcs dispersing across the Fermi level further support HfRhGe as a promising platform for topological superconductivity. Additionally, its noncentrosymmetric nature with time-reversal symmetry breaking superconducting state suggests that it can exhibit an intrinsic superconducting diode effect, offering novel optical and transport properties, with potential applications in dissipationless quantum electronics.