Magnetic monopoles have eluded experimental detection since their prediction nearly a century ago by Dirac. Recently it has been shown that classical analogues of these enigmatic particles occur as excitations out of the topological ground state of a model magnetic system, dipolar spin ice. These quasi-particle excitations do not require a modification of Maxwell's equations, but they do interact via Coulombs law and are of magnetic origin. In this paper we present an experimentally measurable signature of monopole dynamics and show that magnetic relaxation measurements in the spin ice material $Dy_{2}Ti_{2}O_{7}$ can be interpreted entirely in terms of the diffusive motion of monopoles in the grand canonical ensemble, constrained by a network of ``Dirac strings'' filling the quasi-particle vacuum. In a magnetic field the topology of the network prevents charge flow in the steady state, but there is a monopole density gradient near the surface of an open system.