The covariant Gaussian approximation is applied to the U(1) Higgs model. It is shown that the Ward identities are satisfied in this approximation. There is no consistent renormalization with both the kinetic term and bare scalar self-coupling b positive. The theory is trivial in the same sense as the pure 4 theory with a positive b. There exists however a consistent renormalization with a positive b and negative kinetic term. We find in this theory that the Higgs phenomenon does not occur and the photon remains massless.