The experienced keeper will provide Clarets' boss Brian Laws with competition for Brian Jensen.
"In the Premier League you need goalkeeping experience and Nicky has an abundance of it," Burnley boss Brian Laws told the club website.
Weaver started his career at Mansfield Town before joining Manchester City as a teenager, with subsequent spells at Sheffield Wednesday and Charlton.
The former England Under-21 international will join up with the Clarets ahead of their game against Chelsea on Saturday, and is likely to go straight into the squad.
"The chance to come back into the Premier League was just something I couldn't turn down and hopefully it's longer than just a few months," Weaver told the Burnley club website.
"Once I heard Burnley were interested then I pretty much made my mind up and the rest was formalities and I'm delighted to be here."
Weaver is Laws' second recruit during the transfer window, following the signing of Monaco striker Frederic Nimani on a five-month loan deal.
Weaver was great for us (Dundee United) Shetland_James His signing will be a boost for the beleaguered Turf Moor outfit, who have lost all three games since Laws took over from Owen Coyle.
And Laws added: "He has played at the highest level and he is still only 30."
During his spell in Scotland, Weaver earned the praise of former Dundee United manager Craig Levein for his displays, before Levein accepted the Scotland post.
The Yorkshire-born player spent a decade at Manchester City, making more than 200 appearances, although he missed the best part of three years due to a serious knee injury.