  We calculate the 1-loop effective potential of an Abelian Higgs model within
the R_{\xi/\sigma} class of non-linear gauges that preserves the Higgs-boson
low-energy theorem. The R_{\xi/\sigma} gauge involves two gauge-fixing
parameters \xi and \sigma, and is a renormalizable extension of the Feynman--'t
Hooft R_\xi set of gauges beyond the 1-loop level. By taking consistently into
account Goldstone--gauge-boson mixing effects, we show how the 1-loop effective
potential evaluated at its extrema is independent of both \xi and \sigma, in
agreement with the Nielsen identity. The 1-loop constant and wavefunction
renormalizations are presented within this non-linear R_{\xi/\sigma} class of
gauges.
