ok, from one who has experienced such a loss.
Do not minimize anything. Don't say "at least" or "She's not suffering anymore", or any of those things.
Give her a hug. Send her a card, and if you knew the dog, recall a special memory of her. Say you are sorry, and if you have had such a loss, say you understand how it feels.
Offer an ear to listen, or a shoulder for a hug or to cry on if needed.
The loss I experienced was the sharpest of my entire life so far. The support I received from understanding friends, many from very far away, who really did understand helped me get through it.
I honestly thought I would drown under the weight of the grief.
It was the first time in my life I had emotions that were truly uncontrollable. I cried every single day, several times. for months.
I hid it pretty well except from those closest to me.
Please convey to her my sincerest condolences. At the time I did not believe it, but time, (and in time, a new little puppy) healed most of the wound for me.
I truly do know how it feels.