  We consider a Glauber dynamics associated with the Ising model on a large
two-dimensional box with with minus boundary conditions and in the limit of a
vanishing positive external magnetic field. The volume of this box increases
quadratically in the inverse of the magnetic field. We show that at subcritical
temperature and for a large class of starting measures, including measures that
are supported by configurations with macroscopic plus-spin droplets, the system
rapidly relaxes to some metastable equilibrium ---with typical configurations
made of microscopic plus-phase droplets in a sea of minus spins--- before
making a transition at an asymptotically exponential random time towards
equilibrium ---with typical configurations made of microscopic minus-phase
droplets in a sea of plus spins inside a large contour that separates this plus
phase from the boundary. We get this result by bounding from above the local
relaxation times towards metastable and stable equilibria. This makes possible
to give a pathwise description of such a transition, to control the asymptotic
behaviour of the mixing time in terms of soft capacities and to give estimates
of these capacities.
