This paper will be published as part of the Galaxy Evolution Explorer (GALEX) ApJ Letters Special Issue. Links to the full set of papers will be available at http://www.galex.caltech.edu/PUBLICATIONS/ after Nov. 22, 2004. For version with full resolution images go to: http://edoc.mpg.de/204700.0 ; The total ultraviolet (UV) flux (from 1412 to 2718 AA) of M 101 is compared on a pixel-to-pixel basis with the total far-infrared (FIR) flux (from 60 to 170 micron), using the maps of the galaxy taken by GALEX in the near-UV and far-UV and by ISOPHOT at 60, 100 and 170 micron. The main result of this investigation is the discovery of a tight dependence of the FIR/UV ratio on radius, with values monotonically decreasing from around 4 in the nuclear region to nearly zero towards the edge of the optical disk. Although the tightness of this dependence is in part attributable to resolution effects, the result is consistent with the presence of a large-scale distribution of diffuse dust having a face-on optical depth which decreases with radius and which dominates over the more localized variations in opacity between the arm and interarm regions. We also find a trend for the FIR/UV ratio to take on higher values in the regions of diffuse interarm emission than in the spiral-arm regions, at a given radius. This is interpreted quantitatively in terms of the escape probability of UV photons from spiral arms and their subsequent scattering in the interarm regions, and in terms of the larger relative contribution of optical photons to the heating of the dust in the interarm regions.