Place the chopped cucumber in a colander, and sprinkle with 3/4 teaspoon salt. Toss well, and drain for 30 minutes. Place the cucumber on several layers of paper towels, and cover with additional paper towels. Let the cucumber stand for 5 minutes, pressing down occasionally to absorb any excess water.
While cucumber drains, spoon yogurt into a colander lined with a double thickness of paper towels or a coffee filter. Let stand 5 minutes. Scrape into a bowl using a rubber spatula. Stir in cucumber, mint, oil, remaining 1/4 teaspoon salt, pepper, and garlic. Refrigerate, covered, until ready to serve.