We present the first simultaneous measurement of the interaction rates of $pp$, $^7$Be, and $pep$ solar neutrinos performed with a global fit to the Borexino data in an extended energy range (0.19-2.93)$,$MeV. This result was obtained by analyzing 1291.51$,$days of Borexino Phase-II data, collected between December 2011 and May 2016 after an extensive scintillator purification campaign. We find: rate($pp$)$,$=$,$$134$$,$$pm$$,$$10$$,$($stat$)$,$$^{rm +6}_{rm -10}$$,$($sys$)$,$cpd/100$,$t, rate($^7$Be)$,$=$,$$48.3$$,$$pm$$,$$1.1$$,$($stat$)$,$$^{rm +0.4}_{rm -0.7}$$,$($sys$)$,$cpd/100$,$t, and rate($pep$)$,$=$,$$2.43$$pm$$,$$0.36$$,$($stat$)$^{+0.15}_{-0.22}$$,$($sys$)$,$cpd/100$,$t. These numbers are in agreement with and improve the precision of our previous measurements. In particular, the interaction rate of $^7$Be $ u$s is measured with an unprecedented precision of 2.7%, showing that discriminating between the high and low metallicity solar models is now largely dominated by theoretical uncertainties. The absence of $pep$ neutrinos is rejected for the first time at more than 5$,$$sigma$. An upper limit of $8.1$$,$cpd/100$,$t (95%$,$C.L.) on the CNO neutrino rate is obtained by setting an additional constraint on the ratio between the $pp$ and $pep$ neutrino rates in the fit. This limit has the same significance as that obtained by the Borexino Phase-I (currently providing the tightest bound on this component), but is obtained by applying a less stringent constraint on the $pep$ $ u$ flux.
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