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Get one of these.
About £6 uk. You don't say if you're from the UK, but saying coffee grounds I'd guess not. You ought to be ablt to get something like this in the US.
Acts like a caffetiere, there is a mesh filter which travels in a groove on the inside of the cup trapping the grounds at the bottom of the mug. Genius.
Source: my life. I use one every day at work.