Context. Gamma-ray burst (GRB) emission is believed to originate in highly relativistic fireballs. Aims. Currently, only lower limits were securely set to the initial fireball Lorentz factor Γο. We aim to provide a direct measure of Γο. Methods. The early-time afterglow light curve carries information about Γο, which determines the time of the afterglow peak. We have obtained early observations of the near-infrared afterglows of GRB 060418 and GRB 060607A with the REM robotic telescope. Results. For both events, the afterglow peak could be clearly singled out, allowing a firm determination of the fireball Lorentz of Γο ~400, fully confirming the highly relativistic nature of GRB fireballs. The deceleration radius was inferred to be R[dec] ≈ 10¹⁷ cm. This is much larger than the internal shocks radius (believed to power the prompt emission), thus providing further evidence for a different origin of the prompt and afterglow stages of the GRB.