According to The Sun, the pair wed at a town hall in Ravello, near the Amalfi Coast, in a service costing £400 and attended by two witnesses.
Councillor Luigi Apicella, who performed the ten-minute civil ceremony, said: "I didn't know it was the Gary Lineker until someone told me. It was a very normal wedding and there was nothing extravagant.
"He didn't even have a jacket or tie on and they walked to the town hall. The bride wore a cream dress and the service was in Italian with a local girl translating for them."
Lineker and Bux are expected to celebrate their wedding today with a large reception and hold another party this weekend.
Singer Duffy has been tipped to perform at the celebration.