  We show that in the category of complex algebraic varieties, the
Eilenberg--Moore spectral sequence can be endowed with a weight filtration.
This implies that it degenerates if all involved spaces have pure cohomology.
As application, we compute the rational cohomology of an algebraic $G$-variety
$X$ ($G$ being a connected algebraic group) in terms of its equivariant
cohomology provided that $H_G(X)$ is pure. This is the case, for example, if
$X$ is smooth and has only finitely many orbits. We work in the category of
mixed sheaves; therefore our results apply equally to (equivariant)
intersection homology.
