A search for the rare decay ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{e}^{+}{e}^{\ensuremath{-}}$ has been carried out at the Lawrence Berkeley Laboratory using a sparkostrictive wire-chamber spectrometer. Analysis programs identified events with three tracks or with two tracks of which one was an electron and the other a positron. Analysis of the three-track events yields a branching ratio $\frac{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{e}^{+}{e}^{\ensuremath{-}})}{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}\mathrm{all})}l1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ (90% confidence). The two-track events, analyzed by searching for sparks on a possible pion track, yield a limit of $\frac{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{e}^{+}{e}^{\ensuremath{-}})}{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}\mathrm{all})}l2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ (90% confidence). The latter value implies that the coupling constant for a vector neutral current must be about three orders of magnitude smaller than the coupling constant for the charged vector current.