  Given a weak distributive law between algebras underlying two weak
bialgebras, we present sufficient conditions under which the corresponding weak
wreath product algebra becomes a weak bialgebra with respect to the tensor
product coalgebra structure. When the weak bialgebras are weak Hopf algebras,
then the same conditions are shown to imply that the weak wreath product
becomes a weak Hopf algebra, too. Our sufficient conditions are capable to
describe most known examples, (in particular the Drinfel'd double of a weak
Hopf algebra).
