  We show that the sphaleron energy which identifies with the (instanton)
potential barrier for B-violation is reduced in the presence (background) of an
electroweak Z-string. We also show that for large enough Higgs coupling,
$\lambda$, the sphaleron energy can go negative. For such $\lambda$,
electroweak Z-strings can reduce their energy by accumulating sphaleron bound
states. This further endows them with baryon number. Given our approximation,
the value of $\lambda$ at which this occurs is rather large to be realistic for
the standard electroweak model.
