We observe a slowly-evolving quiet region solar eruption of 1999 April 18, using EUV images from the EUV Imaging Telescope (EIT) on the Solar and Heliospheric Observatory (SOHO), and soft X-ray images from the Soft X-ray Telescope (SXT) on Yohkoh. Using difference images, where an early image is subtracted from later images, we examine dimmings and brightenings in the region for evidence of the eruption mechanism. A filament rose slowly at about 1 kilometer per second for six hours before being rapidly ejected at about 16 kilometers per second, leaving flare brightenings and post-flare loops in its wake. Magnetograms from the Michelson Doppler Imager (MDI) on SOHO show that the eruption occurred in a large quadrupo1ar magnetic region, with the filament located on the neutral line of the quadrupole's central inner lobe, between the inner two of the four polarity domains. In step with the slow rise, subtle EIT dimmings commence and gradually increase over the two polarity domains on one side of the filament, i.e. in some of the loops of one of the two side lobes of the quadrupole. Concurrently, soft X-ray brightenings gradually increase in both side lobes. Both of these effects suggest heating in the side-lobe magnetic arcades, which gradually increase over several hours before the fast eruption. Also during the slow pre-eruption phase, SXT dimmings gradually increase in the feet and legs of the central lobe, indicating expansion of the central-lobe magnetic arcade enveloping the filament. During the rapid ejection, these dimmings rapidly grow in darkness and in area, especially in the ends of the sigmoid field that erupts with the filament, and flare brightenings begin underneath the fast-moving but still low-altitude filament. We consider two models for explaining the eruption: breakout, which says that reconnection occurs high above the filament prior to eruption, and tether cutting, which says that the eruption is unleashed by reconnection beneath the filament. The pre-eruption evolution is consistent with gradual breakout that led to (and perhaps caused) the fast eruption. Tether-cutting reconnection below the filament begins early in the rapid ejection, but our data are not complete enough to determine whether this reconnection began early enough to be the cause of the fast-phase onset. Thus, our observations are consistent with gradual breakout reconnection causing the long slow rise of the filament, but allow the cause of the sudden onset of the explosive fast phase to be either a jump in the breakout reconnection rate or the onset of runaway tether-cutting reconnection, or both.