Mark and James - your answers are basically correct but should be clarified. Having a registered service dog is not enough to waive a pet deposit.
If the animal is for emotional support and provides no other service, a landlord may be required to waive their "no pets" policy, though a deposit may be required. The key document to receive an exemption is a doctor's prescription or therapist's letter stating that the animal is required for health reasons.
A "service dog" needs to be required by you to assist with normal living functions, such as a seeing-eye or hearing dog. In some cases emotional assistance animals also qualify as service animals. Service animals are exempt from deposit requirements, though the tenant is responsible for any damage done to the premises by the animal.
My point is that certifying the animal doesn't qualify it as an emotional support or service dog - you must establish that it provides a necessary service. The Americans With Disabilities Act (ADA) does not require that an animal be certified.
Here is a link below to the National Service Animal Registry that answers you questions.