  We study the potential of the LHC and future low-energy experiments to
precisely measure the underlying model parameters of a new Z' boson. We
emphasize the complimentary information obtained from both on- and off-peak LHC
dilepton data, from the future Q-weak measurement of the weak charge of the
proton, and from a proposed measurement of parity violation in low-energy
Moller scattering. We demonstrate the importance of off-peak LHC data and
Q-weak for removing sign degeneracies between Z' couplings that occur if only
on-peak LHC data is studied. A future precision measurement of low-energy
Moller scattering can resolve a scaling degeneracy between quark and lepton
couplings that remains after analyzing LHC dilepton data, permitting an
extraction of the individual Z' couplings rather than combinations of them. We
study how precisely Z' properties can be extracted for LHC integrated
luminosities ranging from a few inverse femtobarns to super-LHC values of an
inverse attobarn. For the several example cases studied with M_Z'=1.5 TeV, we
find that coupling combinations can be determined with relative uncertainties
reaching 30% with 30 fb^-1 of integrated luminosity, while 50% is possible with
10 fb^-1. With SLHC luminosities of 1 ab^-1, we find that products of quark and
lepton couplings can be probed to 10%.
