But even if a Catholic priest officiates, a marriage is not valid if one of the partners has a living spouse.
The priest is a witness, the ministers of the sacrament are the couple, and if one of them is married, that person lacks the capacity to minister the sacrament.
Right, but I understood the question to be whether the Catholic priest could officiate in a subsequent marriage taking place after the annulment. If Kerry's first marriage was nullified, then in the eyes of the church he did not have a living spouse and it would be acceptable to marry him in the church.