Instructions
Go to Lake Verity after meeting with Dialga at Spear Pillar. Surf to the small island cavern in the middle of the lake and enter the cave. Find Mesprit inside and talk to him. He disappears after the meeting, but the encounter is saved to your Pokédex.
Pull out your Pokétch and click the red button until you are looking at the Marking Map application. There will be a symbol flashing on and off on the map; this represents Mesprit's location.
Walk to Mesprit's location as displayed on your Marking Map application. Do not fly; if you do, Mesprit's location will automatically change. Walk around the tall grass or surf in the water below Mesprit's map marker until you engage in a random battle.
Fight Mesprit to weaken him. Use either Mean Look or Block to prevent Mesprit from running away during the first turn. Attack Mesprit with regular attacks and any status effect attacks, such as Sleep Powder and Thunderwave. Mesprit will most likely run away before the battle is over; however, any damage you deliver to him during the fight will be carried over to your next encounter with him.
Continue pursuing Mesprit by using the Map Marker application in your Pokétch, just as you did before. Battle him until he is weak enough for you to capture him with a Quick Ball or an Ultra Ball.