  We present a high-field magnetization study of the $S$ = 1/2
antiferromagnetic Heisenberg chain [PM Cu(NO$_3$)$_2$(H$_2$O)$_2$]$_n$. For
this material, as result of the Dzyaloshinskii-Moriya interaction and a
staggered $g$ tensor, the ground state is characterized by an anisotropic
field-induced spin excitation gap and a staggered magnetization. Our data
reveal the qualitatively different behavior in the directions of maximum and
zero spin excitation gap. The data are analyzed via exact diagonalization of a
linear spin chain with up to 20 sites and on basis of the Bethe ansatz
equations, respectively. For both directions we find very good agreement
between experimental data and theoretical calculations. We extract the magnetic
coupling strength $J/k_B$ along the chain direction to 36.3(5) K and determine
the field dependence of the staggered magnetization component $m_s$.
