Molecules with bistable spin states are widely studied because of their importance to the natural world and their potential applications as molecular scale switches. In molecular crystals and framework materials, elastic interactions between molecules lead to collective phenomena including hysteresis, multistep transitions, and antiferroelastic order of spin states. Elastic frustration, the inability to simultaneously minimize competing elastic interactions, plays a key role in many of the most important phenomena in spin crossover materials. Here we use an elastic model to predict that a new phase of matter occurs for bistable molecules on the kagome lattice, which is intrinsically frustrated as it is composed of equilateral triangles. In this phase, which we call "spin-state ice" in analogy to water and spin ices, there is no long-range order of spin-states; instead they follow a local "ice rule" that each triangle must contain two metal centers in one spin state and one in the other. We show that spin-state ice supports mobile collective excitations that carry a spin midway between the two spin states of a single metal center but no electrical charge. We show that there are distinctive signatures of spin-state ice in neutron scattering, electron paramagnetic resonance, and thermodynamic experiments.