Pulsars are known to power winds of relativistic particles that can produce bright nebulae by interacting with the surrounding medium. These pulsar wind nebulae are observed by their radio, optical, and x-ray emissions, and in some cases also at TeV (teraelectron volt) energies, but the lack of information in the gamma-ray band precludes drawing a comprehensive multiwavelength picture of their phenomenology and emission mechanisms. Using data from the AGILE satellite, we detected the Vela pulsar wind nebula in the energy range from 100 MeV to 3 GeV. This result constrains the particle population responsible for the GeV emission and establishes a class of gamma-ray emitters that could account for a fraction of the unidentified galactic gamma-ray sources.