The differential and total cross sections for the dp--> 3Heeta reaction have been measured in a high precision high statistics COSY-ANKE experiment near threshold using a continuous beam energy ramp up to an excess energy Q of 11.3 MeV with essentially 100% acceptance. The kinematics allowed the mean value of Q to be determined to about 9 keV. Evidence is found for the effects of higher partial waves for Q >or= 4 MeV. The very rapid rise of the total cross section to its maximum value within 0.5 MeV of threshold implies a very large eta3He scattering length and hence the presence of a quasibound state extremely close to threshold.