Hand your spectator a full deck of cards and ask her to examine and shuffle it.
Take the deck back and tell her that you are going to try to memorize the entire deck. Tell her that the world record for memorizing a shuffled deck is 48 seconds but that you feel confident you can do it in less than 10. Quickly, and with a studious expression, rifle through the deck pretending to memorize the cards as you go.
Hand the deck back to your spectator and ask her to pick one card and put it in her pocket. This is a completely free and fair choice on her part. Tell her that to prove your skills, you will look through the deck and identify which card she took.
At this point, you have no idea what card she took. The first thing to do is identify its color. With the deck face up, perform a very quick Russian shuffle (see the references for instructions on how to perform a Russian shuffle). The cards will fly from one hand to the other at lightning speed. Then pretend to have identified the color of your spectator's card. Ask her the following question:
"I think I have identified the color of your card. It's not a black card, is it?"
This is an open question. If she says "Yes it is", say "I thought so" and if she "No it isn't", say "No, I didn't think it was." By using an open question you can respond in a way that makes it seem like you knew the color of her card, regardless of what she says. For this example let's pretend your spectator picked a red card, the 7 of hearts.
To identify which specific card was taken, you need to clock the deck. Clocking is a very simple mathematical technique. Take the deck and thumb through the cards, focusing only on the reds. Add up the values of each card as you go. Whenever your total goes above 10, subtract 10 so that you are left with a 1-digit number.
For example, let's say the first six red cards in the deck are the 4 of Hearts, 8 of Diamonds, 7 of Hearts, 6 of Diamonds, 3 of Hearts and the 9 of Diamonds. Here's how you would clock them:
4+8=12, 12-10=2, 2+7=9, 9+6=15, 15-10=5, 5+3=8, 8+9=17, 17-10=7
Your "clocked" total would be 7. Each time your running total goes above 10, immediately subtract before adding the next card. This sounds difficult, but there are tricks you can use to make it easier. Because you subtract 10 every time your running total goes above 10, you can ignore the 10 of Hearts and 10 of Diamonds completely. The Jack, Queen, and King, are worth 11, 12, and 13 respectively. The easiest way to "clock" them is to just add 1, 2 or 3.
If this sounds difficult, don't worry. With a couple of hours practice you ought to be able to get this down to 15-20 seconds. It's just a matter of training. Remember, you never need to add higher than 10.
Once you've clocked all the red cards in the deck, take your total and subtract it from 12. That will give you the value of your spectator's card. For example, if your total is 5 then you know your spectator is holding a 7.
However, the clocking method is not 100% perfect. If your final result is 1 then the spectator's card could be an Ace (1), or a Jack (11). If it is 2, then the card could be a 2 or a Queen (12). If it is 3 then it could be a 3 or a King (13).
Thumb through the cards a second time. To make this seem more natural say to your spectator "I think I know which card you picked, but let me make sure."
At this point you know the value of the card (or, at the very least, you have narrowed it down to two possible values) so you need to look again to see which red card of that value remains. You can then work out your spectator's exact card. Even if you are unlucky enough to have to search for two cards, say, a 2 or a Queen, the operation is not difficult.
Our spectator chose the 7 of hearts so as soon as we see the 7 of Diamonds, we can finish the trick and tell the spectator her card. This will appear miraculous because she had complete control over which card she chose, she shuffled the deck, and she also checked it for gimmicks. The only conclusion she can draw is that you memorized an entire deck in 10 seconds.