(Abridged) Star formation-driven outflows are a critical phenomenon in theoretical treatments of galaxy evolution, despite the limited ability of observations to trace them across cosmological timescales. If the strongest MgII absorption-line systems detected in the spectra of background quasars arise in such outflows, ultra-strong MgII (USMgII) absorbers would identify significant numbers of galactic winds over a huge baseline in cosmic time, in a manner independent of the luminous properties of the galaxy. To this end, we present the first detailed imaging and spectroscopic study of the fields of two USMgII absorber systems culled from a statistical absorber catalog, with the goal of understanding the physical processes leading to the large velocity spreads that define such systems. Each field contains two bright emission-line galaxies at similar redshift (dv < 300 km/s) to that of the absorption. Lower-limits on their instantaneous star formation rates (SFR) from the observed OII and Hb line fluxes, and stellar masses from spectral template fitting indicate specific SFRs among the highest for their masses at z~0.7. Additionally, their 4000A break and Balmer absorption strengths imply they have undergone recent (~0.01 - 1 Gyr) starbursts. The concomitant presence of two rare phenomena - starbursts and USMgII absorbers - strongly implies a causal connection. We consider these data and USMgII absorbers in general in the context of various popular models, and conclude that galactic outflows are generally necessary to account for the velocity extent of the absorption. We favour starburst driven outflows over tidally-stripped gas from a major interaction which triggered the starburst as the energy source for the majority of systems. Finally, we discuss the implications of these results and speculate on the overall contribution of such systems to the global SFR density at z~0.7.
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