In this paper a partially ordered semigroup (S,\(\cdot,\leq)\) with greatest element e is called weakly commutative if for every a,b\(\in S\) there is \(n\in {\mathbb{N}}\) such that \((ab)^ n\leq bea\). The main result states that every such semigroup S is a semilattice of archimedean subsemigroups (where a subsemigroup T of S is archimedean if for all a,b\(\in T\) there is some \(k\in {\mathbb{N}}\) such tha \(a^ k\leq xby\) for some x,y\(\in T)\). In contrast to the well-known decomposition of commutative semigroups into archimedean subsemigroups, the semilattice congruence on S giving archimedean components need not be unique. It is shown that uniqueness holds iff for every semilattice congruence \(\rho\) on S, \(a\leq b\) in S implies that \(a\rho\) ab. (Reviewer's remark: The note on archimedean congruence classes on the first page is not true; but the proof of the Theorem, part (2), can be modified easily without using the note.)