It is shown that the integration map IP associated with a closed, equicontinuous spectral measure P in a locally convex space X is weakly compact, if and only if, P has finite range. The proof is based on a mixture of techniques from vector-valued integration theory and locally convex algebras. The same characterization is valid if the equicontinuity requirement on P is relaxed, provided now that P is σ-additive for the topology of uniform convergence on bounded sets in X.