We report on an improved measurement of the B8 solar neutrino interaction rate with the Borexino experiment at the Laboratori Nazionali del Gran Sasso. Neutrinos are detected via their elastic scattering on electrons in a large volume of liquid scintillator. The measured rate of scattered electrons above 3 MeV of energy is 0.223-0.016+0.015(stat) -0.006+0.006(syst) cpd/100 t, which corresponds to an observed solar neutrino flux assuming no neutrino flavor conversion of φB8ES=2.57-0.18+0.17(stat) -0.07+0.07(syst)×106 cm-2 s-1. This measurement exploits the active volume of the detector in almost its entirety for the first time, and takes advantage of a reduced radioactive background following the 2011 scintillator purification campaign and of novel analysis tools providing a more precise modeling of the background. Additionally, we set a new limit on the interaction rate of solar hep neutrinos, searched via their elastic scattering on electrons as well as their neutral current-mediated inelastic scattering on carbon, C12(ν,ν′)C∗12 (Eγ=15.1 MeV).