The authors define a solvmanifold as a connected and simply connected solvable Lie group together with a left-invariant metric. Damek-Ricci spaces are examples of solvmanifolds. These spaces appeared as counter-examples for the Lichnerowicz conjecture, namely, that every harmonic Riemannian manifold would be locally isometric to a two-point homogeneous space. No other simply connected examples have been found. In this paper, the authors show that within a large class of solvmanifolds, the Damek-Ricci spaces are the only harmonic spaces.