Atomically uniform Pb films are successfully prepared on Si(111), despite a large lattice mismatch. Angle-resolved photoemission measurements of the electronic structure show layer-resolved quantum well states which can be correlated with dramatic variations in thermal stability. The odd film thicknesses N = 5, 7, and 9 monolayers show sharp quantum well states. The even film thicknesses N = 6 and 8 do not, but are much more stable than the odd film thicknesses. This correlation is discussed in terms of a total energy calculation and Friedel-like oscillations in properties.