Using observations of the corona taken during the total solar eclipses of 2006 March 29 and 2008 August 1 in broadband white light and in narrow bandpass filters centered at Fe X 637.4 nm, Fe XI 789.2 nm, Fe XIII 1074.7 nm, and Fe XIV 530.3 nm, we show that prominences observed off the solar limb are enshrouded in hot plasmas within twisted magnetic structures. These shrouds, which are commonly referred to as cavities in the literature, are clearly distinct from the overlying arch-like structures that form the base of streamers. The existence of these hot shrouds had been predicted by model studies dating back to the early 1970s, with more recent studies implying their association with twisted magnetic flux ropes. The eclipse observations presented here, which cover a temperature range of 0.9 to 2 ×106 K, are the first to resolve the long-standing ambiguity associated with the temperature and magnetic structure of prominence cavities.