We report on an improved measurement of the $^8$B 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 $0.223substack{+0.015 -0.016},(stat),substack{+0.006 -0.006},(syst)$ cpd/100 t, which corresponds to an observed solar neutrino flux assuming no neutrino flavor conversion of $Phisubstack{rm ES ^8rm B}=2.57substack{+0.17 -0.18}(stat)substack{+0.07 -0.07}(syst)times$10$^6$ 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, $^{12}$C($ u, u$)$^{12}$C* ($E_{gamma}$= 15.1 MeV).
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