We study the spectrum of discrete Schrodinger operators with potential given by a primitive invertible substitution sequence (and in fact our results hold for a larger class of potentials). We show this family of operators has a spectrum which is a dynamically defined Cantor set of zero Lebesgue measure. We also show that the Hausdorff dimension of this set depends analytically on the coping constant lambda and tends to 1 as lambda tends to 0. Finally, we show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly.