  We give a detailed stability analysis of the Z-string in the standard
electroweak model. We identify the mode that determines the stability of the
string and numerically map the region of parameter space where the string is
stable. For $\sin^2 \theta_W = 0.23$, we find that the strings are unstable for
a Higgs mass larger than 23GeV. Given the latest constraints on the Higgs mass
from LEP, this shows that, if the standard electroweak model is realized in
Nature, the existing vortex solutions are unstable.
