We present a detailed map of the reddening in a 1.9 degrees x 1.5 degrees section of the Large Magellanic Cloud (LMC), constructed from UBVI photometry of 2069 O and B main sequence stars. We use two reddening-free photometric parameters to determine the line-of-sight reddening to these stars. We find a mean reddening, (E(B-V))(LMC) = 0.20 mag, with a non-Gaussian tail to high values. When the reddening is corrected for foreground Galactic extinction (Oestreicher et al. 1995, A&AS, 112, 495), we find (E(B-V))(LMC) = 0.13 mag. The Line-of-sight values are then interpolated onto a uniform grid with a local least-squares plane fitting routine to construct a reddening map of the region. We use the distribution of reddening values to constrain the line-of-sight geometry of stars and dust in the LMC, and to test and normalize a standard extinction correction for galaxy photometry. We attempt to distinguish between line-of-sight depth effects and structure in the dust distribution as possible causes for the observed differential reddening through this region. We conclude: (1) that our data are consistent with a vertical exponential distribution of stars and dust in the LMC, for which the dust scale height is twice that of the OB stars; (2) that the dust distribution must be non-uniform (clumpy) to account for the full distribution of measured reddening values (i.e., line-of-sight effects alone are insufficient to explain the observed structure); and (3) that the B-band optical depth, tau(B), through the observed region of the LMC is 0.69<tau(B)<0.82. (C) 1997 American Astronomical Society.