Actually I'd say it's more common today for stars to do their own stunts. In the past decade or two, we've had a number of the great Hong Kong fight choreographers come over and train the stars of American movies and TV shows to do their own stunts, in things like Hercules/Xena and The Matrix and Elektra and so forth. Tom Cruise famously did his own climbing stunts outside the tallest skyscraper in the world in Mission: Impossible -- Ghost Protocol. It's actually pretty commonplace to watch a modern movie's making-of features on the DVD and see the actors talk about the weeks or months of fight training they did so they could do their own stuntwork. Audiences today are so critical of imperfection that it's become increasingly necessary for the actors' own faces to be visible in the stunts. So doubles are generally reserved for the really strenuous and dangerous stuff.
These days they can put the star's face into any stunt:
- A star does an aerial stunt wearing a harness attached to nice, thick safety cables. In post production, all the safety equipment is erased via "digital support removal."
- Even safer: a star does a stunt on a green stage three feet off the floor, and along with digital support removal, the whole "dangerous" or high-in-the-air environment is composited in.
- A stuntman does something dangerous, and later the star's face is composited over the image in post production.
- The star does a "high speed" fight or stunt at slow speed, and the film is digitally edited and manipulated to seem like a frantic and dangerous thing was happening. [The fistfights in QUANTUM OF SOLACE are framed so tight and edited so rapid-fire, I literally can't tell which man is Bond. And I know they didn't bother with any real fight moves.]