It is proved that a constrained Willmore immersion of a 2-torus into the conformal 4-sphere

${S}^{4}$ is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in

${\mathbb{R}}^{4}\cong {S}^{4}\setminus \left\{\infty \right\}$ for some point

$\infty \in {S}^{4}$ at infinity. This implies that all constrained Willmore tori in

${S}^{4}$ 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

*N. J. Hitchin’s* approach [J. Differ. Geom. 31, No. 3, 627–710 (1990;

Zbl 0725.58010)] to the study of harmonic tori in

${S}^{3}$.