We present the results of a new search for variable stars in the Local Group dwarf galaxy Leo A, based on deep photometry from the Advanced Camera for Surveys onboard the Hubble Space Telescope. We detected 166 bona fide variables in our field, of which about 60 per cent are new discoveries and 33 candidate variables. Of the confirmed variables, we found 156 Cepheids, but only 10 RR Lyrae stars despite nearly 100 per cent completeness at the magnitude of the horizontal branch. The RR Lyrae stars include seven fundamental and three first-overtone pulsators, with mean periods of 0.636 and 0.366 d, respectively. From their position on the period-luminosity (PL) diagram and light-curve morphology, we classify 91, 58 and 4 Cepheids as fundamental, first-overtone and second-overtone mode Classical Cepheids (CC), respectively, and two as Population II Cepheids. However, due to the low metallicity of Leo A, about 90 per cent of the detected Cepheids have periods shorter than 1.5 d. Comparison with theoretical models indicate that some of the fainter stars classified as CC could be Anomalous Cepheids. We estimate the distance to Leo A using the tip of the red giant branch (TRGB) and various methods based on the photometric and pulsational properties of the Cepheids and RR Lyrae stars. The distances obtained with the TRGB and RR Lyrae stars agree well with each other while that from the Cepheid PL relations is somewhat larger, which may indicate a mild metallicity effect on the luminosity of the short-period Cepheids. Due to its very low metallicity, Leo A thus serves as a valuable calibrator of the metallicity dependences of the variable star luminosities.