  We prove a mixed version of a conjecture of Griffiths: that the closure of
the image of any admissible mixed period map is quasiprojective, with a natural
ample bundle. Specifically, we consider the map from the image of the mixed
period map to the image of the period map of the associated graded. On the one
hand, we show in a precise manner that the parts of this map parametrizing
extension data of non-adjacent-weight pure Hodge structures are quasi-affine.
On the other hand, extensions of adjacent-weight pure polarized Hodge
structures are parametrized by a compact complex torus (the intermediate
Jacobian) equipped with a natural theta bundle which is ample in Griffiths
transverse directions.
  Our proof makes heavy use of o-minimality, and recent work with B. Klingler
associating a $\mathbb{R}_{an,exp}$-definable structure to mixed period domains
and admissible mixed period maps.
