  On a Weierstra{\ss} elliptic surface $X$, we define a `limit' of Bridgeland
stability conditions, denoted $Z^l$-stability, by varying the polarisation in
the definition of Bridgeland stability along a curve in the ample cone of $X$.
We show that a slope stable torsion-free sheaf of positive (twisted) degree or
a slope stable locally free sheaf is taken by a Fourier-Mukai transform on
$D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzero
fiber degree can be modified so that its inverse Fourier-Mukai transform is a
slope semistable torsion-free sheaf.
