Cosmological weak lensing, the gravitational lensing effect of large-scale structure, provides a unique tool to study the large-scale matter distribution and constrain the power spectrum of matter fluctuations free of assumptions on the relation between luminous and dark matter. Due to the weakness of cosmological shear, it can only be detected statistically requiring accurate correction for systematic effects, most importantly the image point-spread-function. Here we present results from our cosmological weak lensing analysis of archival data from the Advanced Camera for Surveys on board the Hubble Space Telescope. Compared to ground-based observations the excellent resolution obtained from space provides a much higher number density of resolved galaxies which can be used to extract the shear signal. This enables us to locally suppress the shape noise induced by the intrinsic ellipticities of galaxies, which is particularly useful for dark matter mapping and constraining the small-scale power spectrum. We have already completed a pilot study using early data from the ACS Parallel Cosmic Shear Survey and the combined GEMS and GOODS mosaic of the Chandra Deep Field South (CDFS). From the low shear signal detected in the GEMS/GOODS data we conclude that this field is subject to strong sampling variance, with an exceptional under-density in its foreground. In a preliminary analysis of the HST/COSMOS Survey we detect residual systematics at small angular scales, whose origin is currently been investigated. Using only B-mode-free scales we find a preliminary estimate for the normalisation of the matter fluctuation power spectrum σ8 = 0.71 ± 0.09 (68% confidence) for a matter density Ωm = 0.24, where the error includes the uncertainties in the redshift distribution, the Hubble constant, and the shear calibration, as well as a Gaussian estimate for cosmic variance.