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We show that a smectic in a disordered medium (e.g., aerogel) exhibitsanomalous elasticity, with the compression modulus B(k) vanishing and the bendmodulus K(k) diverging as k --> 0. In addition, the effective disorder developslong ranged correlations. These divergences are much stronger than those drivenby thermal fluctuations in pure smectics, and are controlled by a zerotemperature glassy fixed point, which we study in an $\epsilon=5-d$ expansion.We discuss the experimental implications of these theoretical predictions.Comment: 4 RevTeX pgs, 1 ps figure; Slightly revised, published versio

Dirt Softens Soap: Anomalous Elasticity of Disordered Smectics