The Bar is the most productive region of the Small Magellanic Cloud in terms of star formation but also the least studied one. In this paper we investigate the star formation history of two fields located in the SW and in the NE portion of the Bar using two independent and well tested procedures applied to the color-magnitude diagrams of their stellar populations resolved by means of deep HST photometry. We find that the Bar experienced a negligible star formation activity in the first few Gyr, followed by a dramatic enhancement from 6 to 4 Gyr ago and a nearly constant activity since then. The two examined fields differ both in the rate of star formation and in the ratio of recent over past activity, but share the very low level of initial activity and its sudden increase around 5 Gyr ago. The striking similarity between the timing of the enhancement and the timing of the major episode in the Large Magellanic Cloud is suggestive of a close encounter triggering star formation. ; Comment: 30 pages, 22 figures, accepted for publication in ApJ