  This paper is the second of the series of papers proposing dedicated
strategies for precision measurements of the Standard Model parameters at the
LHC. The common feature of these strategies is their robustness with respect to
the systematic measurement and modeling error sources. Their impact on the
precision of the measured parameters is reduced using dedicated observables and
dedicated measurement procedures which exploit flexibilities of the collider
and detector running modes. In the present paper we focus our attention on the
measurement of the charge asymmetry of the W-boson mass. This measurement is of
primordial importance for the LHC experimental program, both as a direct test
of the charge-sign-independent coupling of the W-bosons to the matter particles
and as a necessary first step towards the precision measurement of the
charge-averaged W-boson mass. We propose and evaluate the LHC-specific strategy
to measure the mass difference between the positively and negatively charged
W-bosons, MW+ - MW-. We show that its present precision can be improved at the
LHC by a factor of 20. We argue that such a precision is beyond the reach of
the standard measurement and calibration methods imported to the LHC from the
Tevatron program.
