We report the first high spatial resolution submillimeter continuum observations of the Sagittarius B2 cloud complex using the Submillimeter Array (SMA). With the subarcsecond resolution provided by the SMA, the two massive star-forming clumps Sgr B2(N) and Sgr B2(M) are resolved into multiple compact sources. In total, twelve submillimeter cores are identified in the Sgr B2(M) region, while only two components are observed in the Sgr B2(N) clump. The gas mass and column density are estimated from the dust continuum emission. We find that most of the cores have gas masses in excess of 100 M<SUB>&sun;</SUB> and column densities above 10<SUP>25</SUP> cm<SUP>-2</SUP>. The very fragmented appearance of Sgr B2(M), in contrast to the monolithic structure of Sgr B2 (N), suggests that the former is more evolved. The density profile of the Sgr B2(N)-SMA1 core is well fitted by a Plummer density distribution. This would lead one to believe that in the evolutionary sequence of the Sgr B2 cloud complex, a massive star forms first in an homogeneous core, and the rest of the cluster forms subsequently in the then fragmenting structure.