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

Solution 1:

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.