High-quality, stable electron beams are produced from self-injected laser wakefield acceleration using the interaction of moderate 3 TW, 45 fs duration Ti:sapphire laser pulses with high density (>5 × 10 19 cm −3 ) helium gas jet plasma. The electron beam has virtually background-free quasimonoenergetic distribution with energy 35.6 þ3.9 −2.5 MeV, charge 3.8 þ2.8 −1.2 pC, divergence and pointing variation ∼10 mrad. The stable and high quality of the electron beam opens an easy way for applications of the laser wakefield accelerator in the future, particularly due to the widespread availability of sub-10 TW class lasers with a number of laser plasma laboratories around the world.