We show that the real part of the dielectric constant ε′ of rocks at low frequencies can be anomalously high due to the presence of a small concentration η of high aspect ratio particles. For oblate spheroidal grains [Formula: see text] with depolarization factor along symmetry [Formula: see text] axis, [Formula: see text] [Formula: see text] the static value of the dielectric constant of rock [Formula: see text] and dc conductivity [Formula: see text] are given for [Formula: see text] by [Formula: see text] [Formula: see text] and for (2) [Formula: see text] [Formula: see text] Here [Formula: see text] is the dielectric constant of the grain; [Formula: see text] is the dc conductivity of the host rock. Case (1) corresponds to the well known Maxwell‐Wagner effect with [Formula: see text] diverging as η → 0, and [Formula: see text] Case (2) gives a novel result that [Formula: see text] may diverge for [Formula: see text], with a nonvanishing σ(0). Case (2) is applied to explain frequency and salinity dependences and the giant values [Formula: see text] of the dielectric constant of conducting sedimentary rocks. For [Formula: see text], [Formula: see text], [Formula: see text], we find [Formula: see text], which is large compared to [Formula: see text] or the dielectric constant of water [Formula: see text].