Recent intensive Swift monitoring of the Seyfert 1 galaxy NGC 5548 yielded 282 usable epochs over 125 days across six UV/optical bands and the X-rays. This is the densest extended AGN UV/optical continuum sampling ever obtained, with a mean sampling rate <0.5 day. Approximately daily HST UV sampling was also obtained. The UV/optical light curves show strong correlations (r_max = 0.57 - 0.90) and the clearest measurement to date of interband lags. These lags are well-fit by a \tau propto \lambda^4/3 wavelength dependence, with a normalization that indicates an unexpectedly large disk radius of 0.35 +/- 0.05 lt-day at 1367 A, assuming a simple face-on model. The U-band shows a marginally larger lag than expected from the fit and surrounding bands, which could be due to Balmer continuum emission from the broad-line region as suggested by Korista and Goad. The UV/X-ray correlation is weaker (r_max < 0.45) and less consistent over time. This indicates that while Swift is beginning to measure UV/optical lags in general agreement with accretion disk theory (although the derived size is larger than predicted), the relationship with X-ray variability is less well understood. Combining this accretion disk size estimate with those from quasar microlensing studies suggests that AGN disk sizes scale approximately linearly with central black hole mass over a wide range of masses.