  We present a detailed study of the classical Cepheid in the double-lined,
highly eccentric eclipsing binary system OGLE-LMC562.05.9009. The Cepheid is a
fundamental mode pulsator with a period of 2.988 days. The orbital period of
the system is 1550 days. Using spectroscopic data from three 4-8-m telescopes
and photometry spanning 22 years, we were able to derive the dynamical masses
and radii of both stars with exquisite accuracy. Both stars in the system are
very similar in mass, radius and color, but the companion is a stable,
non-pulsating star. The Cepheid is slightly more massive and bigger (M_1 = 3.70
+/- 0.03M_sun, R_1 = 28.6 +/- 0.2R_sun) than its companion (M_2 = 3.60 +/-
0.03M_sun, R_2 = 26.6 +/- 0.2R_sun). Within the observational uncertainties
both stars have the same effective temperature of 6030 +/- 150K. Evolutionary
tracks place both stars inside the classical Cepheid instability strip, but it
is likely that future improved temperature estimates will move the stable giant
companion just beyond the red edge of the instability strip. Within current
observational and theoretical uncertainties, both stars fit on a 205 Myr
isochrone arguing for their common age. From our model, we determine a value of
the projection factor of p = 1.37 +/- 0.07 for the Cepheid in the
OGLE-LMC562.05.9009 system. This is the second Cepheid for which we could
measure its p-factor with high precision directly from the analysis of an
eclipsing binary system, which represents an important contribution towards a
better calibration of Baade-Wesselink methods of distance determination for
Cepheids.
