  The Saffman-Taylor instability occurs when { a} Stokes fluid is displaced by
a less viscous one { in} a Hele-Shaw cell. { This model is useful to study the
secondary oil recovery from a pororus medum.} Since 1960, { polymer solutions}
were used as displacing fluids; moreover, the oil in a porous reservoir can
often be considered a non-Newtonian fluid. Motivated by this fact, in this
paper we study the linear stability of the displacement of two Oldroyd-B fluids
in a rectilinear Hele-Shaw cell, even if the direct relevance for the flow
through porous media is not so evident. We get an { approximate} formula of the
growth rate of perturbations, which depends on the { difference} of the
Weissenberg numbers of the two fluids. { A blow-up of the growth rate appears
for a critical value of this difference.} This singularity is in agreement with
{ previous} numerical and experimental results, already reported by several
papers concerning the flow of complex fluids in Hele-Shaw cells.
