<p>A cohomology of a Lie conformal algebra with coefficients in its conformal module is constructed, which is applied to describe outer conformal derivations and to deduce obstructions of Wells type for conformal derivations to be extensible in abelian extensions of Lie conformal algebras by their conformal modules. Detailed comparisons of this cohomology with the cohomology developed by Bakalov, De Sole, Kac and Voronov are discussed. As examples, some cohomology groups in the case of current conformal algebras are computed.</p>