We report the first observation of the line shape of the Λ(1405) from electroproduction, and show that it is not a simple Breit-Wigner resonance. Electroproduction of K+Λ(1405) off the proton was studied by using data from CLAS at Jefferson Lab in the range 1.0<Q2<3.0 (GeV/c)2. The analysis utilized the decay channels Σ+π- of the Λ(1405) and pπ0 of the Σ+. Neither the standard Particle Data Group resonance parameters, nor free parameters fitting to a single Breit-Wigner resonance represent the line shape. In our fits, the line shape corresponds approximately to predictions of a two-pole meson-baryon picture of the Λ(1405), with a lower mass pole near 1368 MeV/c2 and a higher mass pole near 1423 MeV/c2. Furthermore, with increasing photon virtuality the mass distribution shifts toward the higher mass pole.