  Branching processes in a random environment are natural generalisations of
Galton-Watson processes. In this paper we analyse the asymptotic decay of the
survival probability for a sequence of slightly supercritical branching
processes in an iid random environment, where the offspring expectation
converges from above to $1$. We prove that Haldane's asymptotics, known from
classical Galton-Watson processes, turns up again in the random environment
case, provided that one stays away from the critical/subcritical regime. A
central building block is a connection to and a limit theorem for perpetuities
with asymptotically vanishing interest rates.
