We estimate and remove the contamination of weak gravitational lensing measurements by the intrinsic alignment of close pairs of galaxies. We do this by investigating both the aperture mass B mode statistic, and the shear correlations of close and distant pairs of galaxies. These can be used to quantify non-lensing effects in weak lensing surveys. We reanalyse the {COMBO-17} survey, and study published results from the Red-sequence Cluster Survey and the {VIRMOS-DESCART} survey, concluding that the intrinsic alignment effect is at the lower end of the range of theoretical predictions. We also revisit this theoretical issue, and show that misalignment of baryon and halo angular momenta may be an important effect which can reduce the intrinsic ellipticity correlations estimated from numerical simulations to the level that we and the {SuperCOSMOS} survey observe. We re-examine the cosmological parameter estimation from the {COMBO-17} survey, using the shear correlation function, and now marginalizing over the Hubble constant. Assuming no evolution in galaxy clustering, and marginalizing over the intrinsic alignment signal, we find the mass clustering amplitude is reduced by 0.03 to ??8(??m/0.27)0.6= 0.71 +/- 0.11, where ??m is the matter density parameter. We consider the forthcoming {SuperNova/Acceleration} Probe {(SNAP)} wide weak lensing survey, and the {Canada-France-Hawaii} Telescope Legacy wide synoptic survey; we expect them to be contaminated on scales {\textgreater}1 arcmin by intrinsic alignments at the level of {\textasciitilde}1 and {\textasciitilde}2 per cent, respectively. Division of the {SNAP} survey for lensing tomography significantly increases the contamination in the lowest redshift bin to {\textasciitilde}7 per cent and possibly higher. Removal of the intrinsic alignment effect by the downweighting of nearby galaxy pairs will therefore be vital for {SNAP.}
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