The technically-gifted midfielder has selected the shirt number 11 for his new club and is available to coach Thomas Schneider immediately.
Gruezo, who is currently captain of Ecuador's under-20 team, has already pulled on a Stuttgart shirt having taken part in a trial for the Bundesliga club during their winter training camp in Cape Town.
"We've had Carlos in our sights for quite a while now," said Stuttgart's sporting director Fredi Bobic. "During our training camp in South Africa, he showed what qualities he has. Carlos is a talented player with enormous potential, and he fits the bill perfectly for what we were after."