A semiring is said to be idempotent if both reducts are bands. Let \(\mathbf I\) be the variety of all idempotent semirings, let \(\mathbf D\) be the variety of all distributive lattices. Let \({\mathbf R}^+\) be the subvariety of \(\mathbf I\) which satisfies \(x+y+x=x\). In this paper the authors show that the Mal'cev product \({\mathbf R}^+\circ{\mathbf D}\) forms a variety and consists exactly of the idempotent semirings for which the least lattice congruence is the least lattice congruence on the additive reduct. They specify several systems of identities which determine this variety. Further, section 2 gives other characterizations for \({\mathbf R}^+\circ{\mathbf D}\) and its members. Section 3 gives structure theorems for the semirings of \({\mathbf R}^+\circ{\mathbf D}\) for which the additive reduct is a normal band or a regular band. Section 4 looks at semirings of \({\mathbf R}^+\circ{\mathbf D}\) which can be written as a subdirect product with one of the factors a distributive lattice. Let \(\mathbf{ID}\) be the variety of all idempotent distributive semirings (i.e. \(x+yz=(x+y)(x+z)\), \(xy+z=(x+z)(y+z)\)). The authors show that \(\mathbf{ID}\cap{\mathbf R}^+\circ{\mathbf D}={\mathbf R}^+\vee{\mathbf D}\) consists of the semirings that are a subdirect product of a distributive lattice and a member of \({\mathbf R}^+\).