Excitation of unbalanced-Bessel beams by a gradual increase of nonlinearity in a water sample outlines the achievement of the first ever observed quasimonochromatic wave packet that propagates stably for hundreds of Rayleigh lengths in a focusing and dispersive Kerr medium, i.e., in the absence of spectral broadening and conical emission. A modulational instability analysis reveals the key role of nonlinear dissipation in quenching the growth of spatiotemporal unstable modes.