We present observations of the Type Ia supernova 2003du and report the detectionof an unusual, high-velocity component in the Ca II infrared triplet, similar tofeatures previously observed in SN 2000cx and SN 2001el. This feature exhibits a large expansion velocity (~18,000 km/s) which is nearly constant between -7 and +2 days relative to maximum light, and disappears shortly thereafter. Otherthan this feature, the spectral evolution and light curve resemble those of a normal SN Ia. We find that the Ca II feature can plausibly be caused by a dense shell formed when circumstellar material of solar abundance is overrun by the rapidly expanding outermost layers of the SN ejecta. Model calculations show that the optical and infrared spectra are remarkably unaffected by the circumstellar interaction. In particular, no hydrogen lines are detectable in either absorption or emission. The only qualitatively different features are the strong, high-velocity feature in the Ca II IR-triplet, and a somewhat weaker O I feature near 7,300 AA. The morphology and time evolution of these features provide an estimate for the amount of accumulated matter and an indication of the mixing in the dense shell. We apply these diagnostic tools to SN 2003du and infer that about 2 x 10^{-2} M_sun of solar abundance material may have accumulated in a circumstellar shell prior to the observations. Furthermore, the early light curve data imply that the circumstellar material was originally very close to the progenitor system, perhaps from an accretion disk, Roche lobe or common envelope.
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