une vraie histoire -- a real story (quite a story)
une histoire vraie -- a true story

But that doesn't answer the question. Or are we supposed to assume the answer is "no"?

The question whether the story is actually true or not is not a language question. I don't really know. I tried to clarify what I thought pertained to learning French. Maybe you can answer it?

As a native English speaker, I can say that "quite a story" is sometimes used to imply that a story is false. That is a language question. Probably only a native French speaker can say whether that is also true for "une vraie histoire". 

Yes, "quite a story" can mean that it is either not true or that it is an unexpected or really remarkable story. So it is still ambiguous, even in English.

The native speaker I polled on this (Alsacienne) said that une vraie histoire can be both, just like in English. To her, it could mean either that it was a remarkable story or a spun tale.

I wouldn't say it's ambiguous, exactly. It always means a remarkable story, but in certain situations it clearly implies that you thought it was untrue. But is this also the case in French? Maybe it's never used in this sarcastic sense. We don't know, which is why Lorie asked the question. What does your friend mean by "a spun tale"? Of course "une vraie histoire" can be fictional, but that's not the question. We're talking specifically about the case where someone presents a story as true, and the listener uses the phrase to cast doubt on it. Is that what she means?

In English it is ambiguous. Depending on your tone of voice and context it can mean either. And that seems to be born out in French as well. That's why I gave the English translation in my first response the way I did.

If that's really what you meant by your first response, a few more words would have been helpful.

Well, thanks to you, now there are a lot of words in this response and the answer made clear. :)