We perform an explicit two-loop computation of the Higgs mass in a five-dimensional supersymmetric theory compactified on the orbifold S^1/Z_2, with superpotential localized on a fixed-point brane and supersymmetry broken in the bulk of the extra dimension by Scherk-Schwarz boundary conditions. At one-loop, the Higgs mass is finite and proportional to 1/R^2, R being the radius of the extra dimension. The two-loop corrections contain finite proportional to 1/R^2 and linearly divergent Lambda/R terms, where Lambda is the cutoff of the five-dimensional theory. The divergent terms are cancelled by diagrams containing one-loop wave-function renormalization counterterms, i.e. by linear threshold effects of Yukawa (and gauge) couplings, at the scale 1/R. As a result there is no counterterm for the Higgs mass, which is finite and calculable to the considered order.Work supported in part by CICYT, Spain, under contract AEN98-0816, and by EU under contracts HPRN-CT-2000-00152 and HPRN-CT-2000-00148.Peer reviewed