Right away. I had my shep/rottie mix for three years before I brought home a 10 week old chow mix puppy. My three year old had been extensively socialized since he was a puppy. Had puppy play groups, dog parks, dog classes, brought in a cat, took in foster dogs, he went everywhere with me. This dog was just great...and he had a great affinity for small dogs, puppies, cats. And he was well under my control. So I had no qualms about letting him around my puppy. They were both males and my three year old was neutered. They were fantastic together until my shep/rottie passed away eight years later. Never one fight or growl between them. They wrestled, played, chased, humped, put their mouths on each other...but never aggressively.
I think a lot depends on the temperament of your dog and puppy. My dog was socialized, very patient and was of medium drive. My puppy was a chow mix so more relaxed/aloof. They just fit temperament wise. Most dogs seem to get that it's just a puppy. But I would watch them closely and see how they react to each other.
How is your adult dog's temperament? How is your dog with other dogs? Have you had the dog since it was a puppy? My three year old...let's just say I knew every thought in his head. LOL I could read him since I had him since he was 9 weeks old. I was very confident of his behavior and tolerance levels.