The 31-year-old was a free agent after being released by Aston Villa over the summer.
"I am very happy to sign for such a big club," he told the Hammers' official website.
"I am so pleased to be here and am looking forward to being part of a good team here that will bring this great club back up to the Premier League."
The Norwegian's late registration means he is not eligible to play in West Ham's league opener against Cardiff City.
Carew's former clubs include Rosenborg, Valencia, Lyon and Stoke City, where he spent the second half of last season on loan from Villa.