Using photospheric line-of-sight magnetograms from the National Solar Observatory (NSO) Kitt Peak and coronal X-ray images from the Yohkoh Soft X-Ray Telescope (SXT), we have determined the value of the constant α of the linear force-free field model (∇ ×img1.gif = αimg2.gif) that gives the best visual fit to the overall coronal X-ray structure (αc) of 34 flare-productive active regions of relatively simple bipolar morphology. Vector magnetograms for 24 of these active regions are available from the Haleakala Stokes Polarimeter at Mees Solar Observatory. For each of them, we determine the single best-fit value of α in the photosphere (αp) by three quite different methods and show that these methods give statistically consistent values. By combining this data set with that of the NSO and SXT, we are able to compare for the first time quantitatively and statistically the observed values of α in the photosphere and corona of these regions. We find that the distribution of αp and αc values is fully consistent with the hypothesis that the overall twist density of the magnetic fields of active regions, as measured by the linear force-free field parameter α, is the same in the photosphere and the corona. We therefore conclude that the electric currents that create the nonpotential structure of such solar coronal active regions are of subphotospheric origin and pass without significant modification through the photosphere.