A model for a two-dimensional lipid bilayer, in which both short-range repulsive forces and long-range attractive forces piay a role, is solved exactly. First, in the absence of attractive forces, the configurational entropy is calculated by restricting the lipid hydrocarbon tails to a lattice and solving the resulting counting problem exactly. When long-range attractive forces which decrease exponentially with distance are acting between the polar head groups of the lipids, the partition function can still be calculated exactly, using integration over Gaussian random fields. The thermodynamic functions show a singularity which reflects the completion of the process of self-assemblage of the lipid bilayer. Finally, the model is used to test an approximation method for lipid bilayers.