We present sensitive aperture synthesis observations of the nearby, late-type spiral galaxy NGC 6503, and produce H I maps of considerably higher quality than previous observations by van Moorsel and Wells. We find that the velocity field, while remarkably regular, contains clear evidence for irregularities. The H I is distributed over an area much larger than the optical image of the galaxy, with spiral features in the outer parts and localized holes within the H I distribution. The absence of absorption toward the nearby quasar 1748+700 yields an upper limit of 5 × 10¹⁷ cm^–2 for the column density of cold H I gas along a line of sight which should intersect the disk at a radius of 29 kpc. This suggests that the radial extent of the H I disk is not much larger than that which we trace in H I emission (23 kpc). The observed H I distribution is inconsistent with models of a single thin or thick disk. Instead, the data require a model containing a thin disk plus a thicker low column-density H I layer that rotates more slowly than the thin disk and that extends only to approximately the optical radius. This suggests that the presence of extraplanar gas in this galaxy is largely the result of star formation in the disk rather than cold gas accretion. Improved techniques for interferometric imaging including multi-scale Clean that were used in this work are also described.