Unlocking Pal Park
To unlock the Pal Park feature, you must first have seen all Pokemon in the Sinnoh Pokedex and have obtained the National Pokedex from Professor Oak by speaking with Professor Rowan. After you've done so, travel from Route 220 to Route 221 using Surf, then travel to its end to reach the Pal Park. Although you can reach the Pal Park even before obtaining the National Pokedex, you'll be denied access until you've done so.
Migrating Rayquaza
Once you have reached Pal Park, save your game and switch off your Nintendo DS. Insert the "Pokemon Emerald" cartridge into the Game Boy Advance slot on your Nintendo DS and launch the "Pokemon Emerald" game. Transfer Rayquaza and five other Pokemon into your PC, then save your game and switch off your Nintendo DS.
Launch the "Pokemon Diamond" game and select "Migrate from Emerald," then select Rayquaza and the five other Pokemon you want to migrate. Once you've selected all six Pokemon, a confirmation screen appears. Press "Yes" to save both games, removing the six Pokemon from your "Pokemon Emerald" save and migrating them to your "Pokemon Diamond" save.
Catching Rayquaza
Load the "Pokemon Diamond" save and speak with the man inside Pal Park to begin a Catching Show. You will be given six Park Balls, special Pokeballs with a 100 percent capture rate, and be allowed inside the Pal Park catching area. Run around in the grassy area in the bottom left corner until you encounter Rayquaza, then select "Throw Ball" from the menu to catch it. Catch the other five Pokemon you migrated as well to automatically leave the Pal Park catching area. You will be given the option to put Rayquaza and the other five Pokemon into your PC.
Restrictions
To migrate Rayquaza and other Pokemon from "Pokemon Emerald" to "Pokemon Diamond," you need a Nintendo DS or Nintendo DS Lite console, since the Nintendo DSi doesn't have a Game Boy Advance slot. Additionally, you will be unable to migrate Rayquaza if he knows a Hidden Machine move; the game will alert you if this is the case. You can unlearn the move by visiting the Move Deleter in Lilycove City.