In this paper, an inf-sup stable Galerkin mixed finite element method, for the equations of motion arising in the 2D Oldroyd model of order one, with a grad-div stabilization is discussed and an error analysis is carried out. Optimal error estimates for the velocity in L ∞(L 2 )-norm and for the pressure in L 2 (L 2 )-norm are established in the semidiscrete case. Then, based on backward Euler method, a completely discrete scheme is analyzed, and optimal error estimates are derived. This grad-div stabilization scheme, which adds a stabilization term to the momentum equation, is known to be suitable for high Reynolds number. In conformation with this, the error estimates, in both semi and fully discrete cases, are obtained with constants independent of inverse of viscosity. Finally, we present some numerical results to validate our theoretical results.