We give counterexamples for the open problem, posed by T.D. Blackmore (1997), of whether weak amenability of a semigroup algebra l^1(S) implies complete regularity of the semigroup S. We present a neat set of conditions on a commutative semigroup (involving concepts well known to those working with semigroups, e.g. the counterexamples are nil and 0-cancellative) which ensure that S is irregular (in fact, has no nontrivial regular subsemigroup), but l^1(S) is weakly amenable. Examples are then given.