I would think such an animal would be a constant companion .

There is not yet much case law on service/companion animals.Make sure you document real non-discriminatory reasons why you do not want to rent to the first applicant.  

They meet other successful people, they form friendships and forge partnerships: companions, collaborations and more.Sometimes, a shift in paradigm can return big profits in terms of results and income!

There are two key issues here.1) As REI pointed out, if the doctor "prescribes" a companion animal, then a "no Pets" rule is irrelevant, as this is now a "service" animal issue, which DOES bring the ADA into play.2) More to the point however, the OP is speaking of an "apartment", AND states that "other" people have pets.

Dogs are not a protected class unless they are a service or companion animal, and then essentially they are treated like another person, they are no longer a pet. 

I got a short typed paragraph from a medical professional stating "It is my clinical judgement that this individual would benefit from the allowance of a pet companion at the residence."  

A pet doesn't need to be a trained service animal, many are therapy/companion animals.

A tenant had lived in the building for several years a 1st floor unit,, she had bad accident.. she loved living there, the family hired a contractor to put in everything she needed to assist in her daily living,, including a cement ramp to enter and exit the building... she was disabled, wheelchair bound, yet able to transfer and mentally able to act on her own behalf, and was also able to have a dog, due to companion needs.. 

The Alabama Senate has a companion Bill (SB261) that is waiting on the call of the chair to bring the Bill to a vote.

In our student house we use a separate Utilities Budget agreement as a companion to the lease.