  Weak gravitational lensing is a powerful tool for studying both the geometry
and the dynamics of the Universe. Its power spectrum contains information on
the sources emitting photons and on the large--scale structures that these
hotons cross. We calculate the weak lensing cosmic convergence and shear power
spectra, in linear theory and Limber's approximation, for two different classes
of cosmological models: the standard \LambdaCDM and Unified Dark Matter (UDM)
models. The latter models attempt to solve the problems of the dark matter in
the dynamics of galaxies and galaxy clusters and of the late-time acceleration
of the Universe expansion by introducing a scalar field that mimics both dark
matter and dark energy. A crucial feature of the UDM models is the speed of
sound c_\infty, that is the value of the sound speed at late times, on which
structure formation depends. In this paper, we provide the predictions of the
UDM models for the weak lensing signal,with various values of c_\infty. We
consider both the Cosmic Microwave Background and background galaxies at
different redshifts as sources for the weak lensing power spectra. We find that
the power spectra in UDM models are more sensitive to the variations of
c_\infty for sources located at low redshifts. Moreover, we find that for
c_\infty>10^{-3} (in units of the speed of light), the UDM weak lensing
convergence power spectrum C^{\kappa\kappa}(l) for background galaxies is
strongly suppressed with respect to the \LambdaCDM spectrum, particularly at
small angular scales l\gtrsim100.
