Unambiguous evidence for the microscopic coexistence of ferromagnetism and superconductivity in UCoGe ($T_{rm Curie} sim 2.5$ K and $T_{rm SC}$ $sim$ 0.6 K) is reported from $^{59}$Co nuclear quadrupole resonance (NQR). The $^{59}$Co-NQR signal below 1 K indicates ferromagnetism throughout the sample volume, while nuclear spin-lattice relaxation rate $1/T_1$ in the ferromagnetic (FM) phase decreases below $T_{rm SC}$ due to the opening of the superconducting(SC) gap. The SC state was found to be inhomogeneous, suggestive of a self-induced vortex state, potentially realizable in a FM superconductor. In addition, the $^{59}$Co-NQR spectrum around $T_{rm Curie}$ show that the FM transition in UCoGe possesses a first-order character, which is consistent with the theoretical prediction that the low-temperature FM transition in itinerant magnets is generically of first-order.
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