  We explicitly construct families of integrable $\sigma$-model actions
smoothly interpolating between exact CFTs. In the ultraviolet the theory is the
direct product of two current algebras at levels $k_1$ and $k_2$. In the
infrared and for the case of two deformation matrices the CFT involves a coset
CFT, whereas for a single matrix deformation it is given by the ultraviolet
direct product theories but at levels $k_1$ and $k_2-k_1$. For isotropic
deformations we demonstrate integrability. In this case we also compute the
exact beta-function for the deformation parameters using gravitational methods.
This is shown to coincide with previous results obtained using perturbation
theory and non-perturbative symmetries.
