  We investigate, in a model-independent way, the conditions required to obtain
a satisfactory model of extended inflation in which inflation is brought to an
end by a first-order phase transition. The constraints are that the correct
present strength of the gravitational coupling is obtained, that the present
theory of gravity is satisfactorily close to general relativity, that the
perturbation spectra from inflation are compatible with large scale structure
observations and that the bubble spectrum produced at the phase transition
doesn't conflict with the observed level of microwave background anisotropies.
We demonstrate that these constraints can be summarized in terms of the
behaviour in the conformally related Einstein frame, and can be compactly
illustrated graphically. We confirm the failure of existing models including
the original extended inflation model, and construct models, albeit rather
contrived ones, which satisfy all existing constraints.
