Like many of the achievements in this game, this one can be obtained fairly painlessly in Sandbox mode by making the game as easy as possible. Unlimited money, easy difficulty, etc. Set the win condition to Construction: Space Program.
Now, your first (and really only) order of business is making the U.S. happy. Start by building an embassy and inviting the U.S. Then, Praise them as often as you can. Change your Constitution to stuff that the U.S. likes. Most of them are pretty obvious but in case you're not sure what's what, here's a rundown of what the U.S. will like:
Voting Rights: All Citizens Vote
Religion & State: Secular State
Armed Forces: Professional Army
Political Rights: Democracy
Labor Policy: Happy Childhood
Citizenship: Visa Program
Personal Rights: Open Society
Economy Structure: Free Market
Media Independence: Independent Media
Digital Rights: Fight Piracy
Globalization: Open Borders
Ecology: Energy Efficiency
...Okay, done all that now? Then the U.S. should be head over heels (or it would, if states had heels) in love with you. If you're not at 100, use your Embassy to Praise the U.S. until you are.
Then, simply quickbuild the Space Program (or whatever you set the win condition as) and enjoy your shiny new achievement. This method is easier as it allows you to pick exactly when the game ends because the achievement specifies that you have to finish a game while U.S. relations are at 100.