A nearby normal spiral galaxy, NGC 3627, which is a member of the interacting group the Leo Triplet, is mapped in the CO 1-0 transition for its central 4.5 kpc region using the Berkeley-Illinois-Maryland Association (BIMA) interferometer array with 7" resolution, and in the 21 cm emission of neutral hydrogen for the entire galaxy using the Very Large Array (VLA) with 30" resolution. The combined CO and H I data provide new information, both on the history of the past encounter of NGC 3627 with its companion galaxy NGC 3628 and on the subsequent dynamical evolution of NGC 3627 as a result of this tidal interaction. In particular, the morphological and kinematic information indicates that the gravitational torque experienced by NGC 3627 during the close encounter triggered a sequence of dynamical processes, including the formation of prominent spiral structures, the central concentration of both the stellar and gas mass, the formation of two widely separated and outwardly located inner Lindblad resonances, and the formation of a gaseous bar inside the inner resonance. These processes in coordination allow the continuous and efficient radial mass accretion across the entire galactic disk. The observational result in the current work provides a detailed picture of a nearby interacting galaxy which is very likely in the process of evolving into a nuclear active galaxy. It also suggests one of the possible mechanisms for the formation of successive instabilities in postinteraction galaxies, which could very efficiently channel the interstellar medium into the center of the galaxy to fuel nuclear starburst and Seyfert activities.