A participant takes the deck out of the box and shuffles it. Then they cut the deck and the top card is immediately turned over. It is, for example, the 7 of Spades. The deck is turned face-up and clearly spread to show that there are no duplicates.
The participant opens the envelope and removes the card inside. It is a perfect match - the 7 of Spades.
NO sleight-of-hand required. This literally is as easy as it gets!