Stable operation of a laser-plasma accelerator near the threshold for electron self-injection in the blowout regime has been demonstrated with 25–60 TW, 30 fs laser pulses focused into a 3–4 millimeter length gas jet. Nearly Gaussian shape and high nanosecond contrast of the focused pulse appear to be critically important for controllable, tunable generation of 250–430 MeV electron bunches with a low energy spread, ~ 10 pC charge, a few-mrad divergence and pointing stability, and a vanishingly small low-energy background. The physical nature of the near-threshold behavior is examined using three-dimensional particle-in-cell simulations. Simulations indicate that properly locating the nonlinear focus of the laser pulse within the plasma suppresses continuous injection, thus reducing the low-energy tail of the electron beam.