Gear engagement device for a motor-vehicle gearbox, the device comprising: - a gearbox shaft, - a gear wheel (R1, R2) freely rotatably mounted on said shaft and carrying an auxiliary toothed crown (C1, C2), - a hub rigidly connected in rotation to said shaft, - a gear engagement sleeve connected in rotation to said hub and having an inner toothed crown, said sleeve being axially movable relative to the hub from a neutral position towards a gear engagement position in which said auxiliary toothed crown (C1, C2) carried by said gear wheel (R1, R2) is in engagement with the inner toothed crown of the sleeve, so as to connect in rotation said gear wheel (R1, R2) to said shaft, - a synchronizer ring (Y1, Y2) connected in rotation to said hub, and having an outer toothed crown (D1, D2) axially interposed between the inner toothed crown of the sleeve and the auxiliary toothed crown (C1, C2) of the gear wheel (R1, R2), said synchronizer ring having at least one conical surface (W1, W2) cooperating with a conical portion (X1, X2) rigidly connected to said gear wheel (R1, R2) for synchronized engagement of said inner toothed crown of the sleeve on said auxiliary toothed crown (C1, C2) of the gear wheel (R1, R2) when the sleeve is axially moved towards said gear engagement position, - a driving device for driving the axial movement of the sleeve, characterized in that said device is configured so that in said gear engagement position the conical surface (W1, W2) of said synchronizer ring (Y1, Y2) or a front surface (S3, S4) of said sleeve are pressed axially in friction contact against said conical portion (X1, X2) of said gear wheel (R1, R2) or against a respective engagement surface (R1A, R2A) of the gear wheel (R1, R2), respectively, so that this friction contact tends to brake any relative rotations between said gear wheel (R1, R2) and said synchronizer ring (Y1, Y2) or between said gear wheel (R1, R2) and said sleeve, which relative rotations may take place in transitional conditions of operation of the gearbox which involve inversions or abrupt variations of the transmitted torque, due to a clearance between teeth of the auxiliary toothed crown (C1, C2) of the gear wheel (R1, R2), teeth of the synchronizer ring (Y1, Y2) and teeth of the sleeve. - a gearbox shaft, - a gear wheel (R1, R2) freely rotatably mounted on said shaft and carrying an auxiliary toothed crown (C1, C2), - a hub rigidly connected in rotation to said shaft, - a gear engagement sleeve connected in rotation to said hub and having an inner toothed crown, said sleeve being axially movable relative to the hub from a neutral position towards a gear engagement position in which said auxiliary toothed crown (C1, C2) carried by said gear wheel (R1, R2) is in engagement with the inner toothed crown of the sleeve, so as to connect in rotation said gear wheel (R1, R2) to said shaft, - a synchronizer ring (Y1, Y2) connected in rotation to said hub, and having an outer toothed crown (D1, D2) axially interposed between the inner toothed crown of the sleeve and the auxiliary toothed crown (C1, C2) of the gear wheel (R1, R2), said synchronizer ring having at least one conical surface (W1, W2) cooperating with a conical portion (X1, X2) rigidly connected to said gear wheel (R1, R2) for synchronized engagement of said inner toothed crown of the sleeve on said auxiliary toothed crown (C1, C2) of the gear wheel (R1, R2) when the sleeve is axially moved towards said gear engagement position, - a driving device for driving the axial movement of the sleeve,