The shapes and stability of pendant drops in the presence of an electric field is a classical problem in capillarity. This problem has been studied in great detail by numerous investigators when the drops are either perfect conductors or nonconductors and the surrounding fluid is a nondonductor. In this paper, the axisymmetric equilibrium shapes and stability of a nonconducting drop hanging from a nonconducting nozzle that is immersed in a perfectly conducting ambient fluid, a problem that has heretofore not been considered in the literature, are determined by solving the free boundary problem comprised of the Young-Laplace equation for drop shape and an integral equation for the electric field distribution. Here the free boundary problem is discretized by a hybrid technique in which the Young-Laplace equation is solved by the finite element method and the electrostatic problem is solved by the boundary element method. An electrode in the form of a metal rod or an annulus is placed inside and coaxial with the tube whose tip is located a distance H-1 above or below the tube outlet. The electric field is generated by connecting the rod or the annulus electrode to a source of high voltage at potential P and grounding the ambient conducting fluid that surrounds the drop and the tube. When the force due to surface (interfacial) tension is large compared to those due to gravity and electric field, equilibrium drop shapes are sections of spheres and can be parametrized by a parameter -1 less than or equal to D less than or equal to 1 : D = 0 corresponds to a hemisphere, as D --> 1 the drop approaches a sphere, and as D --> -1 the drop becomes vanishingly small. The results show that equilibrium families of fixed drop volume, or fixed D, lose stability at turning points with respect to the applied potential, where P = P*. Detailed computations reveal the importance of varying the drop size and geometric factors such as the location of the tip of the electrode H-1, electrode thickness W, and, in the case of the annulus electrode, the inner radius of the annulus R(A) on the value of P*. Except for very skinny drops, i.e., D --> -1, the computed profiles of drops at their limits of stability imply that nonconducting drops in a conducting ambient fluid should become unstable by pinching off near the contact line. This finding agrees with recent experiments on such drops, but stands in marked contrast to previous studies on conducting and nonconducting drops immersed in a nonconducting ambient fluid where the drops become unstable by developing conical tips from which a jet issues. Very skinny drops, however, take on a dog-boned appearance at their limits of stability regardless of whether the drop is a conductor or a nonconductor immersed in a nonconducting ambient fluid or a nonconductor immersed in a conducting ambient fluid.