We have analyzed Chandra calibration observations of Betelgeuse (alpha Ori, M2Iab, m_V=0.58, 131 pc) obtained at the aimpoint locations of the HRC-I (8 ks), HRC-S (8 ks), and ACIS-I (5 ks). Betelgeuse is undetected in all the individual observations as well as cumulatively. We derive upper limits to the X-ray count rates and compute the corresponding X-ray flux and luminosity upper limits for coronal plasma that may potentially exist in the atmosphere of Betelgeuse over a range of temperatures, T=0.3-10 MK. We place a flux limit at the telescope of fx ~ 4x10^(-15) ergs s^(-1) cm^(-2) at T=1 MK. The upper limit is lowered by a factor of ~3 at higher temperatures, roughly an order of magnitude lower than that obtained previously. Assuming that the entire stellar surface is active, these fluxes correspond to a surface flux limit that ranges from 30-7000 ergs s^(-1) cm^(-2) at T=1 MK, to ~1 ergs s^(-1) cm^(-2) at higher temperatures, five orders of magnitude below the quiet Sun X-ray surface flux. We discuss the implications of our analysis in the context of models of a buried corona and a pervasive magnetic carpet. We rule out the existence of a solar-like corona on Betelgeuse, but cannot rule out the presence of low-level emission on the scale of coronal holes.