A highly conducting charged drop that is surrounded by a fluid insulator of another density can be levitated by suitably applying a uniform electric field. Axisymmetric equilibrium shapes and stability of the levitated drop are found by solving simultaneously the augmented Young–Laplace equation for surface shape and the Laplace equation for the electric field, together with constraints of fixed drop volume, charge, and center of mass. The means are a method of subdomains, finite element basis functions, and Galerkin’s method of weighted residuals, all facilitated by a large‐scale computer. Shape families of fixed charge are treated systematically by first‐order continuation. Previous analyses by Abbasetal. in 1967 and Abbas and Latham in 1969, in which the shapes of levitated drops are approximated as spheroids, are corrected. The new analysis shows that drops charged to less than the Rayleigh limit lose shape stability at turning points, with respect to external field strength, and that the instability seen in experiments of Doyleetal. in 1964 and others is not a bifurcation to a family of two‐lobed shapes, but rather is a related imperfect bifurcation.