Studying massive binaries in different evolution stages or environments may help us to solve the problem of the evolution of massive binaries. The metallicity in the Small Magellanic Cloud (SMC) is much lower than that in our Milky Way, and binaries in the SMC are rarely studied. OGLE-SMC-ECL-2063 is a short-period early-type binary with a period of 0.d6317643 in the SMC. We use the Wilson-Devinney code to analyze its light curves. The result shows that OGLE-SMC-ECL-2063 is an overcontact binary with a high mass ratio of 0.900 and a fill-out factor of 35.9%. The O - C curves of the period of OGLE-SMC-ECL-2063 show a long-term increase with a cyclic oscillation of amplitude A = 0.00503 d and period P-3 = 14.80 yr. All the evidence above indicates that OGLE-SMC-ECL-2063 is in the Case A mass transfer evolutionary state. The mass transfer rate M-2=-5.67 x 10(-7)M(?) yr(-1) is derived and used to explain the continuous period increase. Because both components of OGLE-SMC-ECL-2063 are early-type stars, the existence of a third body may be the reason for the cyclic change in period. The mass of the third body is derived to be no less than 0.70 M(?)and the orbital separation to be no more than 13.22 au. Combining the result of light-curve analysis, the third body tends to be a low-mass late-type star. Such high-mass-ratio binaries play an important role in the evolution of early-type binaries. Thus, researching OGLE-SMC-ECL-2063 provides the basis for us to study the formation and evolution of early-type contact binaries.