Exclusive photoproduction of ρ0(770) mesons is studied using the H1 detector at the ep collider HERA. A sample of about 900,000 events is used to measure single- and double-differential cross sections for the reaction γp→π+π−Y. Reactions where the proton stays intact (mY=mp) are statistically separated from those where the proton dissociates to a low-mass hadronic system (mp<mY<10 GeV). The double-differential cross sections are measured as a function of the invariant mass mππ of the decay pions and the squared 4-momentum transfer t at the proton vertex. The measurements are presented in various bins of the photon–proton collision energy Wγp. The phase space restrictions are 0.5≤mππ≤2.2 GeV, |t|≤1.5 GeV2, and 20≤Wγp≤80 GeV. Cross section measurements are presented for both elastic and proton-dissociative scattering. The observed cross section dependencies are described by analytic functions. Parametrising the mππ dependence with resonant and non-resonant contributions added at the amplitude level leads to a measurement of the ρ0(770) meson mass and width at mρ=770.8+2.6−2.7 (tot.) MeV and Γρ=151.3+2.7−3.6 (tot.) MeV, respectively. The model is used to extract the ρ0(770) contribution to the π+π− cross sections and measure it as a function of t and Wγp. In a Regge asymptotic limit in which one Regge trajectory α(t) dominates, the intercept α(t=0)=1.0654 +0.0098−0.0067 (tot.) and the slope α′(t=0)=0.233+0.067−0.074 (tot.) GeV−2 of the t dependence are extracted for the case mY=mp.