Screening for sensorineural deafness can be done using the BAER test as early as 6 weeks. Anesthesia is not required; the puppy may be held during the test.
Puppies that are unilaterally deaf (deaf in one ear) will not show the defect in any noticeable way and will make good family pets. These puppies should not be used for breeding. They may produce a higher percentage of deaf puppies and may produce a bilaterally deaf (deaf in both ears) puppy who cannot hear at all.
The Louisiana State University website is a great resource for information on deafness in dogs and cats. They include a description of the BAER test, testing sites and additional topics and resources for dealing with deafness in a pet.