The Joke, Explained:
It is, in fact, somewhat of a British joke.
A Double-Entendre is a statement that can be interpreted in two ways, the obvious one being benign, and the second being sexual in nature.
"To give [a girl] one" is (mostly) Brit slang for a male to have sex with a female.
The woman goes into the bar (typical joke setup) and asks for a Double-Entendre.
The second line IS the double-entendre, to wit:
1. "He gave her one" (i.e., he gave her a double-entendre, as a piece of witty banter)
2. "He gave her one" (i.e., he had sex with her).
It's the recursive meaning of the giving of "one"/"a double entendre" as being the DE itself that's the heart of the joke.