An ultrashort (about 30 fs) petawatt laser pulse focused with a wide focal spot (about 100 mm) in a rarefied plasma (n_0 ~ 10^{17} cm^{−3}) excites a nonlinear plasma wakefield which can accelerate injected electrons up to GeV energies without any pulse channeling. Under these conditions, propagation of the laser pulse with an overcritical power for relativistic self-focusing is almost the same as in vacuum. The nonlinear quasiplane plasma wave, whose amplitude and phase velocity vary along the laser path, effectively traps and accelerates injected electrons with a wide range of initial energies. Electrons accelerated over two Rayleigh lengths (about 8 cm) can gain energies up to 1 Gev. In particular, the electrons trapped from a long (~ 330 fs), nonresonant electron beamlet of 1 MeV particles eventually form a low emittance bunch with energies in the range 900±50 MeV. These conclusions follow from two-dimensional simulations performed in cylindrical geometry by means of the fully relativistic time-averaged particle code WAKE [P. Mora and T. M. Antonsen, Jr., Phys. Rev. E 53, R2068 (1996); Phys. Plasmas 4, 217 (1997)].