  We prove that a constrained Willmore immersion of a 2-torus into the
conformal 4-sphere is either of "finite type", that is, has a spectral curve of
finite genus, or is of "holomorphic type" which means that it is super
conformal or Euclidean minimal with planar ends. This implies that all
constrained Willmore tori in the 4-sphere can be constructed rather explicitly
by methods of complex algebraic geometry. The proof uses quaternionic
holomorphic geometry in combination with integrable systems methods similar to
those of Hitchin's approach to the study of harmonic tori in the 3-sphere.
