  With a current-splitting technique, we calculate the gauge-invariant
initial-state radiation to order \oalf\ with soft-photon exponentiation for on-
and off-shell $W$-pair production. This result generalizes the convolution
formula, which is known from the description of the $Z$ resonance, to the case
of the production of two $W$-bosons. After up to eightfold analytical
integrations, a sufficiently smooth integral over three invariant masses
remains to be treated numerically. Including the Coulomb singularity, the
largest corrections are covered. We discuss the corrections in a large energy
range up to $\sqrt{s}$=1~TeV and draw numerical conclusions on their influence
on the $W$-mass determination at LEP~200. Table~1 and figure~5 are revised
after correcting the analytical formulae for the nonuniversal corrections.
