The He II transverse proximity effect-enhanced He II Lyαtransmission in a background sightline caused by the ionizing radiation of a foreground quasar-offers a unique opportunity to probe the morphology of quasar-driven He II reionization. We conduct a comprehensive spectroscopic survey to find z∼3 quasars in the foreground of 22 background quasar sightlines with Hubble Space Telescope/COS He II Lyαtransmission spectra. With our two-tiered survey strategy, consisting of a deep pencil-beam survey and a shallow wide-field survey, we discover 131 new quasars, which we complement with known SDSS/BOSS quasars in our fields. Using a restricted sample of 66 foreground quasars with inferred He II photoionization rates greater than the expected UV background at these redshifts (ΓQSOHe II 5×10–16 s–1) we perform the first statistical analysis of the He II transverse proximity effect. Our results show qualitative evidence for a large object-to-object variance: among the four foreground quasars with the highest ΓQSOHe II only one (previously known) quasar is associated with a significant He II transmission spike. We perform a stacking analysis to average down these fluctuations, and detect an excess in the average He II transmission near the foreground quasars at 3σ significance. This statistical evidence for the transverse proximity effect is corroborated by a clear dependence of the signal strength on ΓQSOHe II. Our detection places a purely geometrical lower limit on the quasar lifetime of tQ 25 Myr. Improved modeling would additionally constrain quasar obscuration and the mean free path of He II-ionizing photons.