Is it possible for a foreigner to master English so well that even the slightliest trace of an accent is not heard in his speech? What can be done in order to do so? Are there trust-worthy ways of studying English to do so?
Yes, but they have to start really early. When I studied child language acquisition (many moons ago), I read something about 8 years of age being the cut-off point. On my CELTA course I was told an even earlier age - about 3.5.
But don't let that discourage you. Everyone can improve - it's just that once the phonemes of your L1 have 'hardened' (a bit like the bones of a child's skeleton) it takes more work to UNlearn what you already know about how to make speech sounds.