Limb loss is not a part of the game, there are no rules for it, and a DM who houserules it in is responsible for deciding what that means and how it is dealt with. Regeneration and the ring thereof are the only references to the idea, and they heal something that cannot happen. It's pure legacy cruft. The third edition of the rules went much more for the style of “fantasy heroics,” and does not want or need to accurately model injury in combat. Combat is the focus of the game and part of the fantasy is that these realistic risks are not risks for our great heroes or dastardly villains.
Furthermore, such a significant change to the rules relative to the default, not to mention the tone and style of game, should be announced ahead of time, as part of the premise of the game. You should already know the answer to your question, because your DM should have explained his homebrew limb-loss system to you before you made characters.
This is doubly true if your DM is using the lack of rules in this area as a “gotcha” response to your playing an undead character, which it certainly seems like he is. Did he tell you he was using houserules that included loss of limbs? Did he warn you that this was a weakness of playing an undead character when you brought the idea up? Because he very much should have, as this risk is not one present in the default game.
If he did, you might be out of luck; I think it’s a pretty poor thing for the game overall, but if you were aware of the risk and accepted that, well, that was the risk you took. I would strongly consider retiring the character at this point, however. A cripple does not make a good adventurer. It does depend on your class and such just how crucial it is, but most classes are going to struggle to the point where continuing an adventuring career would be suicidal.
If there was not warning, I would ask him what solutions are available, ask him out of character to provide a way for my character regain the use of both arms so that we can get on with whatever we were doing before you got crippled. A side quest or something. If this can be resolved, then fine, it's a part of the story and can be fun. I’m all for cool stories.
But with no warning, no communication of the change of rules and tone, and no opportunity to undo the damage, there’s a major problem. Your DM has introduced a massive, permanent, and unannounced change to a player character. If he just expects you to have this massive (and undefined) drawback for the rest of the game, I would consider that a major abuse of DM authority to spring such a large change on a player's character. I would at that point strongly consider leaving the game. This would not be a game I think is worth my time. Not really because the loss of a limb is unplayable necessarily, plus retiring the character is always an option, but this is a sign that this is a DM that oversteps what I consider his bounds to be, and is likely to do it again. I’m not interested into playing in a DM’s power trip.