Katy Perry and Justin Trudeau took their romance public Saturday night ... the lovebirds went out on a date in Paris, France, to celebrate Katy's 45th birthday.

For their first official outing, Katy and Justin took in a cabaret show at the Crazy Horse Paris, where the paparazzi waited in anticipation for them out front.

TMZ obtained exclusive video of what happened next ... After the show, the couple walked out of the theater hand-in-hand, and a fan gave Katy a rose and wished her happy birthday.

The two then walked to a waiting vehicle as the shutterbugs snapped photos of them, before they were driven away. Of course, Katy and Justin looked absolutely stunning together and appeared to be happier than clams. Best Image

Earlier this month, TMZ published photos of the singer and the f

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