We present the results of a search for short-period variable stars in Leo A. We have found 92 candidate variables, including eight candidate RR Lyrae stars. From the RR Lyrae stars, we measure a distance modulus of (m-M)0=24.51+/-0.12, or 0.80+/-0.04Mpc. This discovery of RR Lyrae stars confirms for the first time the presence of an ancient (older than ~11Gyr) population in Leo A, accounting for at least 0.1% of the galaxy's V luminosity. We have also discovered a halo of old (more than ~2Gyr) stars surrounding Leo A, with a scale length roughly 50% larger than that of the dominant young population. We also report the discovery of a large population of Cepheids in Leo A. The median absolute magnitude of our Cepheid sample is M_V_=-1.1, fainter than 96% of SMC and 99% of LMC Cepheids. Their periods are also unusual, with three Cepheids that are deduced to be pulsating in the fundamental mode having periods of under 1 day. Upon examination, these characteristics of the Leo A Cepheid population appear to be a natural extension of the classical Cepheid period-luminosity relations to low metallicity, rather than being indicative of a large population of "anomalous" Cepheids. We demonstrate that the periods and luminosities are consistent with the expected values of low-metallicity blue helium-burning stars (BHeB's), which populate the instability strip at lower luminosities than do higher metallicity BHeB's. Observations of Leo A were obtained at the WIYN 3.5 m telescope on the nights of 20-22 December 2000, using the MIMO camera.