Welnick took over as the cult band's keyboard player in 1990.
"Vince passed from this earth on 2 June 2006... after a decade of battling tragedy while creating beauty and light around him," the website said.
The band formed in the 1960s and were famed for their live shows. They largely retired from performing after founder Jerry Garcia died in 1995.
No cause of death was given on Welnick's website.
Welnick formed his own group, Missing Man Formation, after the Grateful Dead broke up in 2003. He did not take part in any reunions of surviving Grateful Dead members.
"His service to and love for the Grateful Dead were heartfelt and essential. He had a loving soul and a joy in music that we were lucky to share," his band said in a statement.
Welnick is the fourth keyboard player from the Grateful Dead to have died - founding vocalist and keyboard player Ron McKernan died in 1973 of a gastrointestinal haemorrhage, Keith Godchaux died in a car accident in 1980 and Brent Mydland died in 1990 of a drug overdose.
The Grateful Dead epitomised the 1960s California hippy scene - their followers have long nicknamed them The Dead, and are themselves known as Deadheads.
The Grateful Dead had one of the most dedicated followings of the psychedelic rock scene built up around their music and live shows.
Despite their massive live following they notched up only one top 10 hit in the US with Touch of Grey in 1987.