In order to investigate the Higgs mechanism nonperturbatively, we compute the Gaussian effective potential (GEP) of the U(1) Higgs model ("scalar electrodynamics"). We show that the same simple result is obtained in three different formalisms. A general covariant gauge is used, with Landau gauge proving to be optimal. The renormalization generalizes the "autonomous" renormalization for lambda-phi^4 theory and requires a particular relationship between the bare gauge coupling e_B and the bare scalar self- coupling lambda_B. When both couplings are small, then lambda is proportional to e^4 and the scalar/vector mass-squared ratio is of order e^2, as in the classic 1-loop analysis of Coleman and Weinberg. However, as lambda increases, e reaches a maximum value and then decreases, and in this "nonperturbative" regime the Higgs scalar can be much heavier than the vector boson. We compare our results to the autonomously renormalized 1-loop effective potential, finding many similarities. The main phenomenological implication is a Higgs mass of about 2 TeV.