It is demonstrated that synthesizing an ultrahigh-bandwidth, negatively chirped laser pulse by incoherently stacking pulses of different wavelengths makes it possible to optimize the process of electron self-injection in a dense, highly dispersive plasma (n_0 \sim 10^{19} cm^{-3}). Avoiding transformation of the driving pulse into a relativistic optical shock maintains a quasi-monoenergetic electron spectrum through electron dephasing and boosts electron energy far beyond the limits suggested by existing scaling laws. In addition, evolution of the accelerating bucket in a plasma channel is shown to produce a background-free, tunable train of femtosecond-duration, 35 - 100 kA, time-synchronized quasi-monoenergetic electron bunches. The combination of the negative chirp and the channel permits acceleration of electrons beyond 1 GeV in a 3 mm plasma with 1.4 J of laser pulse energy, thus offering the opportunity of high-repetition-rate operation at manageable average laser power.