Towards the end of Avengers: Age of Ultron JARVIS uploads himself into a lifeless Virbranium android. This process creates Vision. Then before the final battle we see Tony Stark uploading a program called FRIDAY into his suit. During the battle we hear a feminine voice in The Iron Man suit instead of JARVIS'. So did JARVIS cease to exist after uploading himself into Vision? If so did JARVIS know that this would happen, or was it because of the Mindstone?
JARVIS was gone, Tony thought... then he found him in pieces, in the internet, protecting the access to the launch codes from Ultron's attempted hack.
Tony re-collected JARVIS and took him back to the tower.
Tony wanted to put JARVIS into the new fancy body to fight against Ultron. I'm guessing he figured that JARVIS wouldn't be quite the same as when he was in the computer alone, due to the stone, but he figured that it would be better than Ultron having it (or destroying it).
Then Quicksilver and Scarlet Witch come in, Quicksilver disconnects everything before the upload is complete and Thor blasts the body with lightning.
When the new entity, Vision, awakes, he says that he's not JARVIS, despite speaking with JARVIS' voice, but that he's not Ultron, either.
I'm guessing that JARVIS' change into Vision is a combination of the Stone, the partially uploaded data (but not personality) from Ultron, and the lightning bolt of Thor.
Now, from a practicality standpoint, I'm guessing that JARVIS, as a standalone entity is "dead" and that FRIDAY (reminiscent of "His Girl Friday") is now going to be the voice of Tony's computer interface. I'm pretty sure Marvel doesn't want to confuse viewers (or listeners, anyway) by having one person (Paul Bettany) play two characters in the same film.
According to the Marvel Wikia, Friday appears as a computer A.I. entity in the Earth-616 series.
It's assumed that JARVIS was uploaded into the Mind Stone, in the same way Ultron was downloaded from it.
Think of Ultron and JARVIS as CPU's and the Mind Stone as the Computer