  We derive a criterion for the breakdown of solutions to the Oldroyd-B model
in $\R^3$ in the limit of zero Reynolds number (creeping flow). If the initial
stress field is in the Sobolev space $H^m$, $m> 5/2$, then either a unique
solution exists within this space indefinitely, or, at the time where the
solution breaks down, the time integral of the $L^\infty$-norm of the stress
tensor must diverge. This result is analogous to the celebrated
Beale-Kato-Majda breakdown criterion for the inviscid Eluer equations of
incompressible fluids.
