We characterize those derivations from the convolution algebra ℓ1(ℤ+) to its dual that are weakly compact, providing explicit examples that are not compact. The characterization is combinatorial, in terms of ‘translation-finite’ subsets of ℤ+, and we investigate how this notion relates to other notions of ‘smallness’ for infinite subsets of ℤ+. In particular, we prove that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.