Exclusive photoproduction of ρ(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<10GeV). 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≤2.2GeV, |t|≤1.5GeV2, and 20≤Wγp≤80GeV. 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 ρ(770) meson mass and width at mρ=770.8-2.7+2.6(tot.)MeV and Γρ=151.3-3.6+2.7(tot.)MeV, respectively. The model is used to extract the ρ(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)=1.0654-0.0067+0.0098(tot.) and the slope α′(t=0)=0.233-0.074+0.067(tot.)GeV-2 of the t dependence are extracted for the case mY= mp.