We search for a systematic way to characterize the low-energy sensitivity of the minimal Weinberg-Salam model to a heavy-Higgs-boson sector. We find that it is convenient to view this theory as the regulated version of a nonlinear σ model coupled to an SU(2)L×U(1) Yang-Mills theory. Within this framework, MH acts as the regulator. Using the symmetry properties of the nonlinear theory, along with a power-counting analysis, we are able to classify low-energy observables according to their sensitivity to MH. We find that, at one loop, the greatest sensitivity is logarithmic. An illustration of these ideas is provided by a calculation of the one-loop, MH-dependent corrections to the natural relation MWMZ θ=1. Finally, we discuss other possible applications of this technique.