Q:

What punctuation should be used in the end of the direct speech before "but"?

There have already been several questions here about how to use punctuation properly in the direct speech. I've looked through most of them and didn't find an answer to my case. So here I go with my case:
What punctuation will be correct in the following sentence:

He did say, "I am going to visit Jack
  today.", but nobody believed him.

I am especially concerned here about the proper punctuation between today and but
EDIT:
To make it more clear I am replacing personal pronounces with a name:

Tom did say, "I am going to visit Jack
  today.", but nobody believed Tom.

A:

I'd never use a full stop there, nor a capital in the beginning. I'd only use end and beginning-of-sentence markers if forced to do so because I was quoting more than one sentence:

He did say, "Today I am going to visit
  Jack. Please do not wait up." But
  nobody believed him.

As you can see, I've cut the sentence up, because I do not think you can have full stops in a quote and then continue the sentence after the quote. This might be a matter of taste; I'm not sure what style book shave to say about this. Two points to consider:

You could use a colon after "he did say". Then you might want to make the first part an independent sentence.
You could put the full stop after "wait up" after the quotation marks.

