  A typical feature of spontaneous collapse models which aim at localizing
wavefunctions in space is the violation of the principle of energy
conservation. In the models proposed in the literature the stochastic field
which is responsible for the localization mechanism causes the momentum to
behave like a Brownian motion, whose larger and larger fluctuations show up as
a steady increase of the energy of the system. In spite of the fact that, in
all situations, such an increase is small and practically undetectable, it is
an undesirable feature that the energy of physical systems is not conserved but
increases constantly in time, diverging for $t \to \infty$. In this paper we
show that this property of collapse models can be modified: we propose a model
of spontaneous wavefunction collapse sharing all most important features of
usual models but such that the energy of isolated systems reaches an asymptotic
finite value instead of increasing with a steady rate.
