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We characterize the ability of the Dark Energy Camera (DECam) to perform relative astrometry across its 500~Mpix, 3 deg^2 science field of view, and across 4 years of operation. This is done using internal comparisons of ~4x10^7 measurements of high-S/N stellar images obtained in repeat visits to fields of moderate stellar density, with the telescope dithered to move the sources around the array. An empirical astrometric model includes terms for: optical distortions; stray electric fields in the CCD detectors; chromatic terms in the instrumental and atmospheric optics; shifts in CCD relative positions of up to ~10 um when the DECam temperature cycles; and low-order distortions to each exposure from changes in atmospheric refraction and telescope alignment. Errors in this astrometric model are dominated by stochastic variations with typical amplitudes of 10-30 mas (in a 30 s exposure) and 5-10 arcmin coherence length, plausibly attributed to Kolmogorov-spectrum atmospheric turbulence. The size of these atmospheric distortions is not closely related to the seeing. Given an astrometric reference catalog at density ~0.7 arcmin^{-2}, e.g. from Gaia, the typical atmospheric distortions can be interpolated to 7 mas RMS accuracy (for 30 s exposures) with 1 arcmin coherence length for residual errors. Remaining detectable error contributors are 2-4 mas RMS from unmodelled stray electric fields in the devices, and another 2-4 mas RMS from focal plane shifts between camera thermal cycles. Thus the astrometric solution for a single DECam exposure is accurate to 3-6 mas (0.02 pixels, or 300 nm) on the focal plane, plus the stochastic atmospheric distortion., Comment: Submitted to PASP