On the basis of electron microscopy studies, Katz and Thompson have concluded that the porosity of some natural sandstones has a fractal structure. In this letter we show that the recent capillary pressure measurements of Melrose also imply a fractal structure. Melrose measured the capillary pressure P_c of Berea sandstone at low saturations S_w of water. His data cannot be rationalized in terms of thin films and nonfractal pendular structures. Instead, the data obey the scaling law S_w = AP_c^D-3 proposed by de Gennes for the partial filling of a fractal structure by a wetting fluid. The fractal dimension D = 2.55 deduced from the capillary pressure data is consistent with the range 2.57 to 2.87 found by Katz and Thompson for the different sandstones they examined. The capillary pressure curve at low saturation of wetting phase thus provides an alternative means of measuring the fractal dimension of the pore space.