We present high resolution (R = 114,000) ultraviolet measurements of the interstellar absorption line profiles of the C IV (1550 Å) high ionization doublet recorded toward the nearby B2Ve star HD 158427 (d ~ 74 pc). These data, which were recorded with the recently re-furbished Space Telescope Imaging Spectrograph instrument on the Hubble Space Telescope, represent the most convincing detection yet of highly ionized C IV absorption that can be associated with interstellar gas located within the boundary of the Local Cavity (LC). Two highly ionized gas clouds at V 1 = -24.3 km s-1 and V 2 = -41.3 km s-1 are revealed in both C IV absorption lines, with the V 1 component almost certainly being due to absorption by the Local Interstellar Cloud (d < 5 pc). Although the observed column densities for both cloud components can be explained by the predictions of current theoretical models of the local interstellar medium, the narrow Doppler width of the V 2 line profile (b = 6.8 km s-1) indicates an unusually low gas temperature of ≤34,000 K for this highly ionized component. It is conjectured that the V 2 cloud may be due to an outflow of highly ionized and hot gas from the nearby Loop I superbubble. These new data also indicate that absorption due to highly ionized gas in the LC can be best described as being "patchy" in nature.