meaning of but

It is a a poorly written sentence, with an ungainly attempt to masquerade as something literary.
I don't know what the writer intended.

A-Stars nailed it. It's pseudo-archaic. It's a failed attempt at "I doubt not I shall soon be well." That "I have no fear but" is modern sub-standard dialectic. The writer used "but" wrong.

Maybe you could use "but" in a different sentence, in well formed one, so we could better understand your question...

A google search reveals that the source of the quote is an Aesop fable. So it's probably more archaic than pseudo-archaic?

http://en.wikisource.org/wiki/Page:An_argosy_of_fables.djvu/51

I don't know the answer to the original question though. Just wanted to point out the

I'm glad that someone gave us this original text, it's a lot easier to understand the sentence... It is definitely a literary expression... Maybe "but" can be understood as "because", to make it more simple:"I have no fear because I'll be better soon" and this also can suggest annoyance, because of this "Do you be good enough to leave me", like the character wanted to say "Don't worry about me, I'