  We use weak lensing data from the Hubble Space Telescope COSMOS survey to
measure the second- and third-moments of the cosmic shear field, estimated from
about 450,000 galaxies with average redshift <z> ~ 1.3. We measure two- and
three-point shear statistics using a tree-code, dividing the signal in E, B and
mixed components. We present a detection of the third-order moment of the
aperture mass statistic and verify that the measurement is robust against
systematic errors caused by point spread function (PSF) residuals and by the
intrinsic alignments between galaxies. The amplitude of the measured
three-point cosmic shear signal is in very good agreement with the predictions
for a WMAP7 best-fit model, whereas the amplitudes of potential systematics are
consistent with zero. We make use of three sets of large Lambda CDM simulations
to test the accuracy of the cosmological predictions and to estimate the
influence of the cosmology-dependent covariance. We perform a likelihood
analysis using the measurement and find that the Omega_m-sigma_8 degeneracy
direction is well fitted by the relation: sigma_8
(Omega_m/0.30)^(0.49)=0.78+0.11/-0.26. We present the first measurement of a
more generalised three-point shear statistic and find a very good agreement
with the WMAP7 best-fit cosmology. The cosmological interpretation of this
measurement gives sigma_8 (Omega_m/0.30)^(0.46)=0.69 +0.08/-0.14. Furthermore,
the combined likelihood analysis of this measurement with the measurement of
the second order moment of the aperture mass improves the accuracy of the
cosmological constraints, showing the high potential of this combination of
measurements to infer cosmological constraints.
