The observation of an electric dipole moment (EDM) in an atomic system of well-defined angular momentum would be direct evidence for violations of both parity and time-reversal invariances. A search has been made for an EDM in the cesium atom, using an atomic-beam magnetic-resonance technique. A large 25-cps voltage is applied across two parallel metal plates situated between the loops of a Ramsey double-hairpin structure, in an atomic-beam apparatus adjusted to observe "flop-in" transitions. A beam of cesium atoms passes between the plates and is there subject to an electric field of about 5\ifmmode\times\else\texttimes\fi{}${10}^{4}$V/cm. The "flop-in" transition (4, -3) \ensuremath{\leftrightarrow} (4, -4) is observed for a low magnetic field with the rf oscillator adjusted so that the signal at the detector corresponds to the point of maximum slope on either side of the central peak of the Ramsey pattern. If the atom possessed an EDM, it would produce a 25-cps component in the signal at the detector when 25-cps ac voltage alone is applied to the electric field plates. Using phase-sensitive detection techniques, it is possible to separate out such a component from the expected 50-cps component due to the quadratic Stark interaction. The width at half-intensity of the central Ramsey peak is about 2 kc/sec, and shifts of the order of a few parts in ${10}^{5}$ of this are easily detected. A 25-cps component of the detector signal is observed. However, it is likely that this signal is caused by one or more instrumental effects which can simulate an EDM signal. Such instrumental effects are discussed in detail. Also, measurements are performed on other alkali atoms, and comparisons are made between their signals. The results of these comparisons show that the interaction between the atom's magnetic dipole moment and the magnetic field [$(\frac{\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}}{c})\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}$] due to the atom's motion through the electric field is responsible for most of the linear signals. On the basis of these comparisons and subject to the limitations discussed in the text, our limit to the EDM of the Cs atom is $(5.1\ifmmode\pm\else\textpm\fi{}4.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}e$ cm. This result can be used: (1) to set a limit to the EDM of the free electron. If the electron possesses an EDM, then from a relativistic argument it can be shown that the EDM of the Cs atom would be approximately 133 times larger. Using this enhancement factor, our limit to the EDM of the free electron is $(3.8\ifmmode\pm\else\textpm\fi{}3.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}22}e$ cm. (2) to set a limit to the purity of the Cs ground state. Writing the Cs ground state as $|s〉+a|p〉$, we find $\mathrm{Re}a\ensuremath{\le}1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$, which is of the order of magnitude expected from a parity violation in the weak interactions.