The low-temperature magnetic behavior of the double-layered ruthenate Sr 3 Ru 2 O 7 , as grown from a eutectic Sr 2 RuO 4-Sr 3 Ru 2 O 7 system, was investigated via zero-and transverse-field muon-spin rotation. The gradual increase of the muon relaxation rate observed below 2.5 K, even in the absence of applied magnetic fields, indicates the occurrence of a spontaneous breaking of time-reversal symmetry. The onset of the latter at a temperature above 1.5 K, the T c of the single phase Sr 2 RuO 4 , provides evidence about an unconventional superconducting state in the eutectic phase, which most likely takes place at the interface between the Sr 2 RuO 4 and Sr 3 Ru 2 O 7 domains, or even inside the Sr 3 Ru 2 O 7 phase. We show that the superconducting state manifests a two-component behavior in the transverse-field response with changeover at about T = 2.5 K and T = 1.5 K. The comparison with zero-field μSR data in the Ru-Sr 2 RuO 4 eutectic system rules out the possibility of spurious effects due to embedded Ru islands.