  By running the prospective high-energy e^+ e^- collider TESLA in the GigaZ
mode on the Z resonance, experiments can be performed on the basis of more than
10^9 Z events. They will allow the measurement of the effective electroweak
mixing angle to an accuracy of approximately +- 10^-5. Likewise the W boson
mass is expected to be measurable with an error of about 6 MeV near the W^+ W^-
threshold. In this note, we study the accuracy with which the Higgs boson mass
can be determined from loop corrections to these observables in the Standard
Model. The comparison with a directly observed Higgs boson may be used to
constrain new physics scales affecting the virtual loops. We also study
constraints on the heavy Higgs particles predicted in the Minimal
Supersymmetric Standard Model, which are very difficult to observe directly for
large masses. Similarly, it is possible to constrain the mass of the heavy
scalar top particle.
