In this article we report experimental results on the deformation and the critical breakup conditions of a single drop in a medium under simple shear flow. The role played by both drop and matrix elasticities is quantified by using constant viscosity elastic (Boger) fluids. The experiments were conducted using two transparent parallel disks mounted on a R-18 Weissenberg rheogoniometer. The critical shear rate was determined by imposing successive small changes in shear rate from lower to higher values until the drop breakup was observed. The results show remarkable differences in the mode of deformation and breakup for Newtonian and elastic fluid systems. It is also found that the drop resistance to deformation and breakup increases with increasing elasticity ratio. The contribution of the drop and matrix elasticities is quantified by using an empirical relation established between the drop deformation and the capillary number, Ca. The critical breakup conditions, such as a dimensionless breakup time, t(b)*, and a critical capillary number, Ca-c, are determined as a function of the drop/matrix elasticity ratio, k＇. The values of Ca, and t(b)* are found to increase with increasing k＇.