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You're going to need to buy another copy. The DS doesn't have a hard drive on which game saves are stored, since that's inside the cartridge itself. The DSi has a hard drive for downloadable games and pictures, but it doesn't store game data. DS cards are designed to work on all DS models, therefore the game save is within them.
Keep in mind that you're going to need two DSes to use this data via Memory Link. This is Serebii's explanation: The Memory Link feature allows for your Pokémon Black2 & White2 games to collect data from Black & White related to your game. This is done via either local wireless in the Unova Link feature or over WiFi if you have used the Game Sync feature on both games. Various things are unlocked with this feature and this page is to detail all of these features including where they are unlocked.
If you've used/linked/Game Sync'd/Whatever with Pokemon Dream World at least once with your White 1, then your Game Sync data should be on your profile page on the Pokemon Global Link website. Hope that helps!
I've never owned more than one DS at the same time, so I managed to do this with just my 3DS :)
