Give her the dog, if you like, go to the pound and adopt another dog that otherwise would be "put to sleep", and get on with life. Do not treat this like a "real" legal question.
The above is general legal and business analysis. It is not "legal advise" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.
If you can prove that the dog was a gift to you, I don't think your girlfriend has any further rights, and it sounds like it would be prudent to make a clean break with her. It may require an attorney letter to get your ex- to take the matter seriously and leave your alone, but if you have evidence to support that the dog is in fact yours, an attorney should be able to put together an appropriate, powerful letter in a relatively short amount of time.
That being said, unless you have a gift card or another writing that establishes she gave the dog to you, or have one or more unbiased witnesses who will say she clearly gave the dog to you, she could take the position that she bought the dog for both of you and continue to fight with you in this matter -- which could become draining in terms of time, money and emotions.
To help you in this matter, I would suggest that you put together a list of facts that could help to establish that the dog is yours (e.g., a dog license in your name, a veterinary account in your name, bills paid for the upkeep of the dog using your funds, etc.) as well as a list of facts that would suggest the dog was both of yours or even hers (the document transferring title of the dog to her when she bought it, any amounts paid for the upkeep of the dog by her after she gave the dog to you, etc.), and then consult with an Animal Law attorney in your area (if you can find one).
If you are not able to an Animal Attorney in your local area, you might want to take advantage of the free 10-minute consultations offered by the attorneys @ Canine Custody (second link below). Attorney DuPar is licensed to practice law in California, and can likely provide you will helpful advice.
Also, although some of the links on the site are no longer operational, I have also included a list of laws that are generally applicable to all dogs (not just dachshunds!) in San Diego County (see the first link for further details). As a dog owner, you should be aware of the many laws that apply to dogs and their owners.
All the best to you and your pooch!
Any answer or other information posted above is general in nature and is not intended, nor should it be construed, as legal advice. This posting does not create an attorney-client relationship between you and the posting attorney, and you are urged to engage a qualified attorney who is licensed to practice in the relevant jurisdiction.
The dog was a gift, the dog is yours, and you do not have to share the dog.
This is not legal advice but practical: tell her if she wants the dog she is going to have to sue you and you will counter-sue her for half of all your expenses in caring for the dog, including a daily boarding fee, and then STOP communicating with her. Seriously - just stop.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.