No, the game does not end. Capture 60 different Pokémons and get your PokéDex evaluated by Dr. Oak. He will then upgrade your dex. Then talk to Celio on Island One, he will ask you to get the ruby and sapphire. Ruby can be found in Mt. Ember (That is also where you get Moltres). You'll get the sapphire in the Dotted Hole, but the scientist will take it with him to the Rocket "something". Defeat the scientist, take the sapphire, talk to Celio and THERE! you are now authorized to trade with Pokémon Ruby and Sapphire!
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you must catch all Pokemon and defeat the elite 4.
After you beat your rival will take you to the hall of fame room once your Pokemon are all registered in the hall of fame the game will save itself turn off the game and reopen your file you will end up in pallet town.
The battle will end and the Ho-Oh will disappear. However, it will come back after you defeat the elite 4 again.
Go to Indigo Plateu. You will battle Elite Four there and that is the end of the game. Dont worry, you will still be able to continue with game after you beat them. Train your Pokemon before going to Elite Four because they are strong.
If you ever played any of the other Pokemon games you'd know the ending is when you defeat the Elite Four, become the Champion and enter the Hall of Fame. Everything after that is basically extra and you can do whatever you want.