The process pp--> W + J/psi provides a powerful probe of the production mechanism of charmonium in hadronic collisions, and is also sensitive to multiple parton interactions in the colliding protons. Using the 2011 ATLAS dataset of 4.5 fb-1 of sqrt{s} = 7 TeV pp collisions at the LHC, the first observation is made of the production of W + prompt J/psi events in hadronic collisions, using W-->mu+nu and J/psi-->mu+mu. A yield of 27.4+7.5-6.5 W + prompt J/psi events is observed, with a statistical significance of 5.1 sigma. The production rate as a ratio to the inclusive W boson production rate is measured, and the double parton scattering contribution to the cross section is estimated.