We report on an improved measurement of the $^{8}\mathrm{B}$ 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 ${0.223}_{\ensuremath{-}0.016}^{+0.015}(\mathrm{stat}){\text{ }}_{\ensuremath{-}0.006}^{+0.006}(\mathrm{syst})\text{ }\text{ }\mathrm{cpd}/100\text{ }\mathrm{t}$, which corresponds to an observed solar neutrino flux assuming no neutrino flavor conversion of ${\mathrm{\ensuremath{\Phi}}}_{^{8}\mathrm{B}}^{\mathrm{ES}}=2.5{7}_{\ensuremath{-}0.18}^{+0.17}(\mathrm{stat}){\text{ }}_{\ensuremath{-}0.07}^{+0.07}(\mathrm{syst})\ifmmode\times\else\texttimes\fi{}{10}^{6}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}2}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$. 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, $^{12}\mathrm{C}(\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}})^{12}{\mathrm{C}}^{*}$ (${E}_{\ensuremath{\gamma}}=15.1\text{ }\text{ }\mathrm{MeV}$).