From X-ray images in the ROSAT public archives, we determine soft X-ray fluxes, or flux upper limits, for 74 A-type stars, which have been observed during deep integrations with the PSPC. Nine supposedly single, late A stars (0.20 < B-V < 0.35) are found to coincide with X-ray sources. The X-ray luminosities we infer for these stars range from levels comparable to the Active Sun, at log L_x ∼27.6, to much brighter emission levels similar to those observed for active late-type binary systems, near log L_x ∼30.1. Another 10 sources are identified with early A stars (0.0 < B-V < 0.2). Five of these are confirmed double stars, the rest are ostensibly single. The maximum luminosity we detect in the early A stars, log L_x = 30.1, is 3.5 orders of magnitude brighter than the X-ray upper limits for the nondetected stars. Additional study, including radial velocity monitoring and/or optical interferometry, will be needed to determine whether the putatively single X-ray emitting stars are in fact single, or whether their emission is produced entirely or in part by unknown or unresolved binary companions. The level of X-ray emission associated with chemically normal, single A stars thus far appears to be uncorrelated with any obvious stellar property, including the rotation rate, which is known to greatly influence the dynamo activity and the X-ray emission levels of lower mass stars.