  We analyse the structure of the quotient $\mathrm{A}_\sim(\Gamma,X,\mu)$ of
the space of measure-preserving actions of a countable discrete group by the
relation of weak equivalence. This space carries a natural operation of convex
combination. We show that the convex structure of
$\mathrm{A}_\sim(\Gamma,X,\mu)$ is compatible with the topology, and as a
consequence deduce that $\mathrm{A}_\sim(\Gamma,X,\mu)$ is path connected.
Using ideas of Tucker-Drob we are able to give a complete description of the
topological and convex structure of $\mathrm{A}_\sim(\Gamma,X,\mu)$ for
amenable $\Gamma$ by identifying it with the simplex of invariant random
subgroups. In particular we conclude that $\mathrm{A}_\sim(\Gamma,X,\mu)$ can
be represented as a compact convex subset of a Banach space if and only if
$\Gamma$ is amenable. We consider the space $\mathrm{A}_{\sim_s}(\Gamma,X,\mu)$
of stable weak equivalence classes and show that is always a compact convex
subset of a Banach space. For a free group $\mathbb{F}_N$, we show that if one
restricts to the compact convex set $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,\mu)
\subseteq \mathrm{A}_{\sim_s}(\mathbb{F}_N,X,\mu)$ of the stable weak
equivalence classes of free actions, the extreme points are dense in
$\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,\mu)$.
