McCartney plays keyboard on the simple, poignant song that is written as a letter from West's mother, Donda, who died in 2007.
The song is meant to be a message to West's young daughter, North.
West's wife, Kim Kardashian West, praised the song on Twitter, telling fans, "People always ask me what my favorite Kanye song is and it's Only One. Kanye feels like his mom sang thru him to our daughter."
The song concludes with the repeated lyric, "Tell Nori about me." Nori is West's daughter's nickname.
The single, which was released on West's website and through iTunes, is the first song from what has been described by West's reps as "a prolific musical collaboration between these two legendary artists."
The song is from West's forthcoming untitled album, his first since "Yeezus" in 2013.
West and McCartney first began collaborating at a bungalow in Los Angeles in early 2014. The song "Only One" came out of a brainstorming session between the two, with McCartney improvising on the keyboard and West creating lyrics in a stream of consciousness style.
According to the release from West, "Kanye sat there with his family, holding his daughter North on his lap, and listened to his vocals, singing, "Hello, my only one ..." And in that moment, not only could he not recall having sung those words, but he realized that perhaps the words had never really come from him."
Another rumored collaboration between West and McCartney, a song titled "Piss On My Grave" was leaked (no pun intended) earlier this year on Instagram.
The bit of song is no longer available online.