To get to Mauville, you need to do a couple things.
1) Attempt to drop off the parts at the shipyard by talking to the man standing in front of the blueprints. He will say that you should go find Capt. Stern.
2) Now, go to the museum. Notice there are no longer any Aqua members blocking the entrance. Walk inside. But, before you go upstairs, talk to the Aqua Grunt closest to the door and he will give you TM49 (Theif). NOW head upstairs.
3) A couple of Aqua's will attempt to steal your package after talking to Stern, but once you beat them, he will take the parts, and the road to Mauville will be open. Just head North out of the city until you reach the Trick House. Once there, turn to your right. This is the road to Mauville.
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the third badge in sapphire is at mauville city(north of slateport city)
From Mauville City, go south, you'll meet the Fisherman. Surf and keep to east. Then, you reach the New Mauville.
You fly to Mauville City and then you go left.
You can find it in the city called Mauville City the big purple building.
Defeat the 3rd Gym Leader, Wattson, in Mauville City.