This is called shooting "day for night" -- and it's very hard to make it look truly real. You can darken or increase the contrast in post-production -- but shooting it right will be critical.
The best ways to make it work:
1) Show as little sky as possible. Daytime sky is a dead giveaway.
2) Try to shoot right after sunset or right before sunrise ("magic hour"), before the sun is completely up.
3) If your camcorder has manual white balance, try tinting your shot bluish. You can also do this in post-production.
If you can shoot at night, even better. Just light people from the sides rather than from directly in front.
I would also add a blue tint to the footage.
One of the things that will be missing will be the artificial lights, which would be on at night. When Hollywood does "day for night," they often use a bunch of very bright visible lights, so get some sort of indication that the artificial lights are on, and the angle, as Steve mentions, will be very, very important. The daytime shadows will also be a give away, as one would only have softer shadows from the artificial lights, and would have one for each source.
If you can find it, American Cinematographer (last month, or so), has an article on the set for Boardwalk, where they talk a bit about doing day for night shots on that set. Now, in their case, they also built a monster cyc to Key out and insert night time shots of Atlantic City.