We observe a thermomechanical effect when a chemical-potential difference is created by a temperature difference imposed between two liquid reservoirs connected to each other through Vycor rods in series with solid hcp 4He. By creating a temperature difference, ΔT, between the two reservoirs, we induce a rate-limited growth of a pressure difference between the two reservoirs, ΔP. In equilibrium ΔP vs ΔT is in quantitative agreement with the thermomechanical effect in superfluid helium. These observations confirm that below ∼600 mK a flux-limited flow exists through the solid helium.