An exact solution is presented for Ising-like transitions in a decorated lattice model of a porous medium. The model is solved by decimation of the spins, leading to a space-filling lattice with renormalized parameters. The critical temperature is found to vary as 1/lnL, where L is the number of sites between intersections of the spin chains. Some of the critical exponents differ from those of the ordinary Ising problem. We have also studied the case of a single, infinitely long pore, using both exact and approximate methods. An exploration of finite-width effects reveals surprisingly small (quantitative) deviations from mean-field theory.