  We consider first passage percolation on the Erd\H{o}s-R\'{e}nyi graph with
$n$ vertices in which each pair of distinct vertices is connected independently
by an edge with probability $\lambda/n$ for some $\lambda>1$. The edges of the
graph are given non-negative i.i.d. weights with a non-degenerate distribution
such that the probability of zero is not too large. We consider the paths with
small total weight between two distinct typical vertices and analyse the joint
behaviour of the numbers of edges on such paths, the so-called hopcounts, and
the total weights of these paths. For $n\to\infty$, we show that, after a
suitable transformation, the pairs of hopcounts and total weights of these
paths converge in distribution to a Cox process, i.e., a Poisson process with a
random intensity measure. The random intensity measure is controlled by two
independent random variables, whose distribution is the solution of a
distributional fixed point equation and is related to branching processes. For
non-arithmetic and arithmetic edge weight distributions we observe different
behaviour. In particular, we derive the limiting distribution for the minimal
total weight and the corresponding hopcount(s). Our results generalise earlier
work of Bhamidi, van der Hofstad and Hooghiemstra, who assume edge weights are
exponentially distributed. The main tool we employ is the method of moments.
