The main asteroid belt (MB) is low in mass but dynamically excited, with much larger eccentricities and inclinations than the planets. In recent years, the Grand Tack model has been the predominant model capable of reconciling the formation of the terrestrial planets with a depleted but excited MB. Despite this success, the Grand Tack is still not generally accepted because of uncertainties in orbital migration. It was recently proposed that chaotic early evolution of Jupiter and Saturn could excite the initially cold MB. However, hydrodynamical simulations predict that the giant planets should generally emerge from the gas disk phase on orbits characterized by resonant and regular motion. Here we propose a new mechanism to excite the MB during the giant planets' ('Nice model') instability, which is expected to have included repeated close encounters between Jupiter and one or more ice giants ('Jumping Jupiter' -- JJ). We show that when Jupiter temporarily reaches a high enough level of excitation, both in eccentricity and inclination, it induces strong forced vectors of eccentricity and inclination within the MB region. Because during the JJ instability Jupiter's orbit 'jumps' around, forced vectors keep changing both in magnitude and phase throughout the whole MB region. The entire cold primordial MB can thus be excited as a natural outcome of the JJ instability. Furthermore, we show that the subsequent evolution of the Solar System is capable of reshaping the resultant MB to its present day orbital state, and that a strong mass depletion is always associated to the JJ instability phase.< Réduire