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 sin2theta(W) = 0.23, we find that the string
s are unstable for a Higgs mass larger than 23 GeV. Given the latest c
onstraints on the Higgs mass from LEP, this shows that, if the standar
d electroweak model is realized in nature, the existing vortex solutio
ns are unstable.