We observe a slow-onset quiet-region filament eruption with the Extreme Ultraviolet Imaging Telescope (EIT) on the Solar Heliospheric Observatory (SOHO) and the Soft X-ray Telescope (SXT) on Yohkoh. This event occurred on 1999 April 18 and was likely the origin of a coronal mass ejection detected by SOHO at 08:30 UT on that day. In the EIT observation, one-half of the filament shows two stages of evolution: stage 1 is a slow, roughly constant upward movement at approximately 1 km/s lasting approximately 0.5 hr, and stage 2 is a rapid upward eruption at approximately 16 km/s occurring just before the filament disappears into interplanetary space. The other half of the filament shows little motion along the line of sight during the time of stage 1 but erupts along with the rest of the filament during stage 2. There is no obvious emission from the filament in the SXT observation until stage 2; at that time, an arcade of EUV and soft X-ray loops forms first at the central location of the filament and then expands outward along the length of the filament channel. A plot of EUV intensity versus time of the central portion of the filament (where the postflare loops initially form) shows a flat profile during stage 1 and a rapid upturn after the start of stage 2. This light curve is delayed from what would be expected if 'tether-cutting' reconnection in the core of the erupting region were responsible for the initiation of the eruption. Rather, these observations suggest that a loss of stability of the magnetic field holding the filament initiates the eruption, with reconnection in the core region occurring only as a by-product.