  We investigate both numerically and analytically the behaviour of a spin-1
antiferromagnetic (AFM) isotropic Heisenberg chain in an external magnetic
field. Extensive DMRG studies of chains up to N=80 sites extend previous
analyses and exhibit the well known phenomenon of the closure of the Haldane
gap at a lower critical field H_c1. We obtain an estimate of the gap below
H_c1. Above the lower critical field, when the correlation functions exhibit
algebraic decay, we obtain the critical exponent as a function of the net
magnetization as well as the magnetization curve up to the saturation (upper
critical) field H_c2. We argue that, despite the fact that the SO(3) symmetry
of the model is explicitly broken by the field, the Haldane phase of the model
is still well described by an SO(3) nonlinear sigma-model. A mean-field theory
is developed for the latter and its predictions are compared with those of the
numerical analysis and with the existing literature.
