At present many new fluids not obeying the Newtonian laws are being studied by scientists because of their industrial importance [1]. The constitutive equations of these fluids have been given by different researchers, for example, Oldroyd [2] and Walters [3]. Beard and Walters [4] solved the two-dimensional boundary layer flow for an elasticoviscous fluid, near a stagnation point for the case of Walter's liquid B″, when the elasticoviscous parameter is very small. A new method is now proposed for determination of the two-dimensional boundary layer flow near a stagnation point for Walter's liquid B″ I, which is valid not only for very small positive elasticoviscous parameters but also for positive elasticoviscous parameters.