We have measured the magnetic penetration depth, λ( T), of YBCO films from optimal to several underdoped states. Films do not degrade under repeated de- and re-oxygenation. Our deoxygenation procedure permits very small incremental reductions in oxygen content. The two-coil measurement apparatus accurately detects small changes in λ(0). At moderate underdoping, the low- T slope of λ −2 is essentially independent of doping. The data are well described by a simple model in which the pseudogapped portions of the Fermi surface do not contribute to electronic properties, and the pseudogapped portions grow with underdoping.