  We determine exactly the phase structure of a chiral magnet in one spatial
dimension with the Dzyaloshinskii-Moriya (DM) interaction and a potential that
is a function of the third component of the magnetization vector, $n_3$, with a
Zeeman (linear with the coefficient $B$) term and an anisotropy (quadratic with
the coefficient $A$) term. For large values of potential parameters $A$ and
$B$, the system is in one of the ferromagnetic phases, whereas it is in the
spiral phase for small values. In the spiral phase we find a continuum of
spiral solutions, which are one-dimensionally modulated solutions with various
periods. The ground state is determined as the spiral solution with the lowest
average energy density. As the phase boundary approaches, the period of the
lowest energy spiral solution diverges, and the spiral solutions become domain
wall solutions with zero energy at the boundary. The energy of then domain wall
solutions is positive in the homogeneous phase region, but is negative in the
spiral phase region, signaling the instability of the homogeneous
(ferromagnetic) state. The order of the phase transition between spiral and
homogeneous phases and between polarized ($n_3=\pm 1$) and canted ($n_3\not=\pm
1$) ferromagnetic phases is found to be second order.
