Siegrist, 30, has been a free agent since leaving Tannadice at the end of the season and comes in to offer back-up to Celtic's No 1 Joe Hart.
“I am really pleased to welcome Ben to Celtic and we look forward to working with him," said Celtic manager Ange Postecoglou. "I have had really positive discussions with him.
"We all saw last season the ability that Ben has, as time and again he delivered fantastic performances.
“He is a really talented keeper with the ambition to always work, to develop himself further and to achieve success and I think in signing Ben we have made a very important addition to our squad.”
Celtic will now look to push through deals for Lanus left-back Alexandro Bernabei and Vitoria Guimaraes midfielder Alfa Semedo plus the permanent transfer of Jota, who was on loan last season from Benfica.
Bernabei has said his goodbyes to team-mates in Argentina and is expected to move to Celtic for a fee in the region of £2.5million, while Portuguese media report that Semedo, from Guinea-Bissau, is set to sign on loan with a mandatory option to buy for potentially £3.5m. Negotiations continue with Jota over a £6.4m transfer, with confidence high that a deal will be struck.
Postecoglou wants the majority of his squad assembled ahead of the players' return to training next week ahead of Celtic's title defence and Champions League campaign.