  The spectrum of cosmological perturbations in the context of the one-bubble
open inflation model is discussed, taking into account fluctuations of the
metric. We find that, quite generically,thin wall single field models have no
supercurvature modes. However, single field models with supercurvature modes do
exist. In these models the density parameter $\Omega$ becomes a random variable
taking a range of values inside of each bubble. We also show that the model
dependence of the continuous spectrum for both scalar and tensor-type
perturbations is small as long as the kinetic energy density of the background
field does not dominate the total energy density. We conclude that the spectrum
of the density perturbation predicted in the single-field model of the
one-bubble open inflation is rather robust. We also consider the spectrum of
scalar and tensor perturbations in a model of the Hawking-Turok type, without a
false vacuum.
