The 34-year-old, who represented England at under-21 level and had stints with Charlton, Dundee United and Burnley, was released on a free transfer by Sheffield Wednesday at the end of last season.
"I'm delighted to get Nicky on board," said Dons boss Derek McInnes.
"To get such an experienced and talented goalkeeper is important and to get him during the pre-season stage is great for him in terms of being able to integrate with the boys.
"We had a few avenues and other options but we feel with the signing of Nicky we've recruited well and Nicky's eagerness and willingness to come here was pivotal."
Weaver has joined up with his new side in Dublin, where they will play three pre-season fixtures in five days against UCD, Malahide and Shelbourne.