Exclusive photoproduction of rho 0(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 gamma p -> pi+pi -Y. Reactions where the proton stays intact (mY=mp) are statistically separated from those where the proton dissociates to a low-mass hadronic system (mpGeV). The double-differential cross sections are measured as a function of the invariant mass m pi pi 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 W gamma p. The phase space restrictions are 0.5 <= m pi pi <= 2.2GeV, |t|<= 1.5GeV2, and 20 <= W gamma p <= 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 m pi pi dependence with resonant and non-resonant contributions added at the amplitude level leads to a measurement of the rho 0(770) meson mass and width at m rho =770.8-2.7+2.6(tot.)MeV and Gamma rho =151.3-3.6+2.7(tot.)MeV, respectively. The model is used to extract the rho 0(770) contribution to the pi+pi- cross sections and measure it as a function of t and W gamma p. In a Regge asymptotic limit in which one Regge trajectory alpha (t) dominates, the intercept alpha (t=0)=1.0654-0.0067+0.0098(tot.) and the slope alpha ' (t=0)=0.233-0.074+0.067(tot.)GeV-2 of the t dependence are extracted for the case mY=mp.