  We study the Monge and Kantorovich transportation problems on
$\mathbb{R}^{\infty}$ within the class of exchangeable measures. With the help
of the de Finetti decomposition theorem the problem is reduced to an
unconstrained optimal transportation problem on the Hilbert space. We find
sufficient conditions for convergence of finite-dimensional approximations to
the Monge solution. The result holds, in particular, under certain analytical
assumptions involving log-concavity of the target measure. As a by-product we
obtain the following result: any uniformly log-concave exchangeable sequence of
random variables is i.i.d.
