This paper investigates the consensus problem for multiagent systems, under the assumptions that the agents are homogeneous and described by a single-input positive state-space model, the mutual interactions are cooperative, and the static state-feedback law that each agent adopts to achieve consensus preserves the positivity of the overall system. Necessary conditions for the problem solvability, which allow us to address only the special case when the state matrix is irreducible, are provided. Under the irreducibility assumption, equivalent sets of sufficient conditions are derived. Special conditions either on the system description or on the Laplacian of the communication graph allow us to obtain necessary and sufficient conditions for the problem solvability. Finally, by exploiting some results about robust stability either of positive systems or of polynomials, further sufficient conditions for the problem solvability are derived. Numerical examples illustrate the proposed results.