The problem of determining the distribution of current in and the impedance of a cylindrical antenna asymmetrically driven by a discontinuity in scalar potential is formulated. An integral equation is derived and solved by the method of successive approximations to obtain general expressions for the current and the impedance. A simple approximate expression for the impedance of the asymmetrically driven antenna involving a series combination of the known impedances of symmetrically driven antennas is obtained. The impedance and the distribution of current for a cylindrical antenna of length 3λ0/4 driven λ0/4 from one end are evaluated. The broad-band properties are discussed. Since the sleeve dipole with its image is equivalent to a super-position of two asymmetrically driven antennas, expressions for the impedance and distribution of current are obtained readily from the current distribution of the asymmetrically driven antenna. Both impedance and current distribution are determined for a sleeve dipole of over-all length 3λ0/4 over a conducting plane driven λ0/4 from the plane. It is shown that this antenna has broad-band properties very superior to those of a conventional dipole.