The first electrician never did get back to me with an estimate. The second electrician that was out here wouldn't even call me back, even after I left a couple messages. What is it with these businesses? I guess they don't really want the work.
So I called out yet another electrician. This one brought a meter and traced where the electrical line ran under ground. They think they even know where the break is, based on the reading. He offered to try and dig it up to just repair the break, but recommended running all new cable through conduit. He said he won't just run a new neutral wire because it ignores possible damage to the other wires (and maybe also because it's not compliant, as Joe ponited out). Waiting on an estimate from him as well.