A spokesman for the Mayday Hospital in Croydon, south London, said: "Michael Finn died on Saturday night at the Mayday Hospital."
His manager Barry Newby said Finn was believed to have died of kidney and liver problems, but the exact cause of death had not been announced yet.
Finn replaced original member Steve "Peregrine" Took in 1970 and stayed for five years.
Toured
He was known among fans for his trademark bongos and played on classic albums such as The Slider.
Marc Bolan was supposed to have said of Finn: "He can't sing... but he looks superb."
Finn left T Rex in 1975 to form his own band and in recent years had toured with his own incarnation of T Rex.
Under the name Mickey Finn's T Rex, he toured extensively around Europe.
He last performed with his band in Germany at the end of December.
Huge following
Formed by flamboyant lead singer Marc Bolan in 1967, T Rex remain best know for hits such as Get it On, Hot Love and Children of the Revolution in the early 1970s.
The band had a huge following in Britain but failed to replicate that success in in the United States and elsewhere.
They are credited with introducing the phenomenon of "glam rock" to pop music and influencing artists such as David Bowie.
The band has been dogged by premature deaths.
Bolan, the most famous member of the band, died when the car he was a passenger in hit a tree in 1977.
Former member Took died through choking on a cherry stone, and Steve Currie (bass player) died in a car crash.