Equations are derived for the isothermal, low-speed fiber spinning of the Oldroyd fluid B. These equations involve three dimensionless groups whose numerical values are restricted to narrow ranges by the boundary conditions that have to be imposed. Asymptotic results are obtained for limiting values of these groups by means of a perturbation analysis. Also, numerical solutions are presented both in terms of velocity profiles and the admissible values of the three dimensionless groups for different fiber draw ratios. These results are discussed in the context of their application to Boger fluids and are also contrasted with the predictions of the Maxwell fluid model. It is found that under appropriate conditions the predictions of the Oldroyd model B can be quite different from those of the upper convected Maxwell model. In particular, the fluid behaves in a Newtonian manner for both very low and very high Deborah number situations.