We show that the real part of the dielectric constant epsilon ' of rocks at low frequencies can be anomalously high due to the presence of a small concentration eta of high aspect ratio particles. For oblate spheroidal grains (a < b = c) with depolarization factor along symmetry (-a) axis, L s congruent to 1 - delta , delta = pi a/2b, the static value of the dielectric constant of rock epsilon s and dc conductivity sigma (0) are given for (1) delta < eta by sigma (0) congruent to delta sigma R /eta , epsilon s congruent to epsilon ' m /eta , and for (2) delta > eta by sigma (0) congruent to sigma R (1 - eta /delta ), epsilon s congruent to eta epsilon ' m /delta 2 . Here epsilon ' m is the dielectric constant of the grain; sigma R is the dc conductivity of the host rock. Case (1) corresponds to the well known Maxwell-Wagner effect with epsilon s diverging as eta --> 0, and sigma (0) --> 0. Case (2) gives a novel result that epsilon s may diverge for delta > eta > delta 2 , with a nonvanishing sigma (0). Case (2) is applied to explain frequency and salinity dependences and the giant values ( approximately 10 4 ) of the dielectric constant of conducting sedimentary rocks. For eta approximately 10 (super -4) , delta approximately 10 (super -3) , epsilon ' m = 10, we find epsilon s approximately 1000, which is large compared to epsilon ' m or the dielectric constant of water epsilon ' w ( approximately 80).