This is the first paper of a series describing our measurement of weak lensing by large-scale structure using archival observations from the Advanced Camera for Surveys (ACS) on board the Hubble Space Telescope (HST). In this work we present results from a pilot study testing the capabilities of the ACS for cosmic shear measurements with early parallel observations and presenting a re-analysis of HST/ACS data from the GEMS survey and the GOODS observations of the Chandra Deep Field South (CDFS). We describe our new correction scheme for the time-dependent ACS PSF based on observations of stellar fields. This is currently the only technique which takes the full time variation of the PSF between individual ACS exposures into account. We estimate that our PSF correction scheme reduces the systematic contribution to the shear correlation functions due to PSF distortions to < 2*10^{-6} for galaxy fields containing at least 10 stars. We perform a number of diagnostic tests indicating that the remaining level of systematics is consistent with zero for the GEMS and GOODS data confirming the success of our PSF correction scheme. For the parallel data we detect a low level of remaining systematics which we interpret to be caused by a lack of sufficient dithering of the data. Combining the shear estimate of the GEMS and GOODS observations using 96 galaxies arcmin^{-2} with the photometric redshift catalogue of the GOODS-MUSIC sample, we determine a local single field estimate for the mass power spectrum normalisation sigma_{8,CDFS}=0.52^{+0.11}_{-0.15} (stat) +/- 0.07 (sys) (68% confidence assuming Gaussian cosmic variance) at fixed Omega_m=0.3 for a LambdaCDM cosmology. We interpret this exceptionally low estimate to be due to a local under-density of the foreground structures in the CDFS.
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