  In this paper we prove that the defocusing, quintic nonlinear Schr\"odinger
initial value problem is globally well-posed and scattering for $u_{0} \in
L^{2}(\mathbf{R})$. To do this, we will prove a frequency localized interaction
Morawetz estimate similar to the estimate made in [11]. Since we are
considering an $L^{2}$ - critical initial value problem we will localize to low
frequencies.
