We exploit the quantum coherence between pair-produced D0 and D0bar in psi(3770) decays to study charm mixing, which is characterized by the parameters x and y, and to make a first determination of the relative strong phase delta between doubly Cabibbo-suppressed D0 -> K+pi- and Cabibbo-favored D0bar -> K+pi-. We analyze a sample of 1.0 million D0D0bar pairs from 281 pb^-1 of e+e- collision data collected with the CLEO-c detector at E_cm = 3.77 GeV. By combining CLEO-c measurements with branching fraction input and time-integrated measurements of R_M = (x^2+y^2)/2 and R_{WS} = Gamma(D0 -> K+pi-)/Gamma(D0bar -> K+pi-) from other experiments, we find cosdelta = 1.03 +0.31-0.17 +- 0.06, where the uncertainties are statistical and systematic, respectively. In addition, by further including external measurements of charm mixing parameters, we obtain an alternate measurement of cosdelta = 1.10 +- 0.35 +- 0.07, as well as xsindelta = (4.4 +2.7-1.8 +- 2.9) x 10^-3 and delta = 22 +11-12 +9-11 degrees.
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