We present the results from a spectroscopic survey for intervening Mg II absorption in the spectra of 381 background QSOs conducted at the MMT telescope. This survey complements our earlier SDSS EDR Mg II survey, extending our results to lower redshift (z 0.15) and weaker Mg II λ2796 rest equivalent width (W 0.1 A). We confirm two major results from that survey: the transition in the W distribution at W ≈ 0.3 A, and the W-dependent evolution of the incidence of systems. The nature of ∂2N/∂z∂W is consistent with the idea that multiple physically distinct components/processes contribute to the incidence of Mg II absorption systems in a W0-dependent manner and evolve at different rates. A significant decrease in the total proper absorption cross section is detected in our MMT data for systems as weak as 1.0 A ≤ W < 1.5 A at z 0.4. We discuss this W0-dependent evolution in the context of the evolution of galaxy structures, processes including superwinds and interactions, and damped-Lyα absorbers. We also consider the possibility that the observed redshift and W dependence of the incidence of absorption in spectroscopic surveys for low-ionization/neutral gas results from the effects of dust-induced extinction.