How do I trap a mouse without killing it?

Try a humane mousetrap.
humane mousetrap - Google Product Search (http://www.google.co.uk/products?hl=en&client=firefox-a&channel=s&rls=org.mozilla:en-GB:official&hs=cGf&resnum=0&q=humane+mousetrap&um=1&ie=UTF-8&sa=X&oi=product_result_group&resnum=1&ct=title)

Supposing you have an ethical reason for not killing mice, what do you plan to do with it after you catch it?

Check out a tin cat available at your local tsc store
It's a small steel box with one way door and a lid
Bait with peanut butter and wear gloves to avoid hantavirus "please read" Homepage | CDC Hantaviruses (http://www.cdc.gov/ncidod/diseases/hanta/hps/)

^^ perhaps making dissections! =D