In this paper we consider a standard Brownian motion in d, starting at 0 and observed until time t. The Brownian motion takes place in the presence of a Poisson random field of traps, whose centers have intensity t and whose shapes are drawn randomly and independently according to a probability distribution , on the set of closed subsets of d, subject to appropriate conditions. The Brownian motion is killed as soon as it hits one of the traps. With the help of a large deviation technique developed in an earlier paper, we find the tail of the probability St that the Brownian motion survives up to time t when where c (0,) is a parameter. This choice of intensity corresponds to a critical scaling. We give a detailed analysis of the rate constant in the tail of St as a function of c, including its limiting behaviour as c or c0. For d3, we find that there are two regimes, depending on the choice of . In one of the regimes there is a collapse transition at a critical value c* (0,), where the optimal survival strategy changes from being diffusive to being subdiffusive. At c*, the slope of the rate constant is discontinuous. For d=2, there is again a collapse transition, but the rate constant is independent of and its slope at c=c* is continuous.