Possible candidates not previously mentioned include (but are not limited to):
Lionel Barrymore
Marlene Dietrich (longer sound career than Garbo)
Emil Jannings
Pola Negri
Erich von Stroheim
Conrad Veidt
I’d add Gary Cooper to that list.
Did silent directors have a lot of success once sound came in? D. W. Griffith struggled.
I guess Alfred Hitchcock did silent movies? He’s so well known for his sound movies, I’m not even sure how big a hit he was before that.
Cecil B. DeMille managed all right…
Should we mention Gloria Swanson?
Wish we still had @Eve with us.
Most of her silent films are long lost, but “Sunset Boulevard” is a classic.
IIRC, she put her career aside for quite a few years to raise her children (and yeah, have an affair with Joseph Kennedy).
Hitchcock was successful with silents, directing ten of them. Though they were shot in England, so he may not have been well known in the states.
John Ford made around 60 silent films. His silent The Iron Horse was a success.
Frank Capra basically invented the character played by Harry Langdon (who was grouped with Chapin, Keaton, and Lloyd as the four great names of silent comedy by James Agee). He directed several of Langdon’s films to great success before sound came in.
King Vidor directed the silent hits The Big Parade and The Crowd and received several Oscar nominations for his sound films.
Of course, it would be easier for a director to make the change, since he would have learned the visual part of directing in the silent days and just had to learn how to direct dialog.
What???
You know, with a gag on. It’s a particular fetish, I understand. 
Just to clarify I meant that Garbo’s sound films were more famous in her lifetime than Chaplin’s but today Chaplin’s one major sound film is more famous. Clearly Chaplin was a more famous star both then and now.
For directors Lubitsch deserves a mention as someone with success with both silent and sound but De Mille is probably the right answer in terms of having mega-hits in both eras.