Exceptional sets for length under restricted families of projections onto lines in R³

It is shown that if A ⊆ R³ is a Borel set of Hausdorff dimension A>1, and if ρθ is orthogonal projection to the line spanned by ( cos θ, sin θ, 1 ), then ρθ(A) has positive length for all θ outside a set of Hausdorff dimension at most 3- A/2.