We prove a T(1)-type theorem which characterizes compactness of singular integral operators whose kernels satisfy a smoothness condition with decay along the diagonal.

In the same spirit of David and Journe's original T(1) Theorem, we provide this characterization in terms of the action of the operator over special families of functions. For this reason, we define a new property of 'weak compactness' and the appropriate substitute for the space BMO.

We apply our main theorem to prove compactness of certain perturbations of the Cauchy Integral.