The electron-positron collider DAPhiNE, the Italian Phi factory, has been recently upgraded in order to implement an innovative collision scheme based on large crossing angle, small beam sizes at the crossing point, and compensation of beam-beam interaction by means of sextupole pairs creating a "crab-waist" configuration in the interaction region. Experimental tests of the novel scheme exhibited an increase by a factor of 3 in the peak luminosity of the collider with respect to the performances reached before the upgrade. In this Letter we present the new collision scheme, discuss its advantages, describe the hardware modifications realized for the upgrade, and report the results of the experimental tests carried out during commissioning of the machine in the new configuration and standard operation for the users.