First of all, I want to make it clear that Amar'e Stoudemire isn't a Knick yet. However, I am sure he will be by the start of the '09-'10 season. Here is why.
Recently, Stoudemire told a Washington D.C. radio station that there was a "50 percent chance" he would return to the Suns next season (NBA.com). This is not a high chance at all. Although he also said he wanted to go to a contender, the Knicks will be a much more improved team in the next season.
This will be so because of the progression of Wilson Chandler, Nate Robinson, and Danilo Gallinari. I also expect the Knicks to draft former Davidson star Stephen Curry, an extremely dangerous shooter whose game is like a combination of Steve Nash and Jamal Crawford.
Another big piece that comes into play is the coach of the Knicks, Mike D'Antoni. D'Antoni coached Amar'e while on the Suns for much of his NBA career, and Stoudemire would fit right into the run and gun system that he knows so well.
Stoudemire will also have a better chance to make it to the playoffs with the Knicks than with the aging Suns. In the '08-'09 playoffs, the Detroit Pistons made the eighth seed with a mediocre 39-43 record, which the Knicks could easily reach or even surpass in the upcoming season if they had Amar'e.
With all of this in play, Amar'e Stoudemire will be a New York Knick during the '09-'10 season.