Finding the bag of faulty coins in one weighing

You have ten bags of coins. Genuine coins weigh $10$ g each, but every coin in exactly one bag is faulty and weighs $1$ g away from $10$ g (either heavier or lighter). Using a digital scale that shows exact weight, how can you find the faulty bag with just one weighing?

Take $k$ coins from bag $k$ for $k=1,\dots,10$ and weigh all $55$ together. Genuine coins would weigh $550$ g, so the reading differs from $550$ by exactly $k$ grams, where $k$ is the faulty bag's number; the sign also tells you heavier or lighter.