The central region of the recently discovered supernova remnant (SNR) RX J0852.0-4622 was observed with the Advanced CCD Imaging Spectrometer detector aboard the Chandra X-Ray Observatory. We found only one relatively bright source, about 4' north of the SNR center, with a flux of ~2 × 10^-12 ergs s^-1 cm^-2 in the 0.5-10 keV band. The position of this pointlike source, CXOU J085201.4-461753, rules out its association with the two bright stars in the field, HD 76060 and Wray 16-30. Observations of the field with the Cerro Tololo Inter-American Observatory 0.9 m telescope show a star (R ≈ 17, B ≈ 19) at about 24 from the nominal X-ray position. We consider association of this star with the X-ray source unlikely and estimate a limiting magnitude of the optical counterpart as B ≥ 22.5 and R ≥ 21.0. Based on the X-ray-to-optical flux ratio, we argue that the X-ray source is likely the compact remnant of the supernova explosion that created the RX J0852.0-4622 SNR. The observed X-ray spectrum of the source is softer than spectra of magnetospheric radiation of rotation-powered pulsars, but it is harder than spectra of cooling neutron stars emitting thermal radiation from the entire surface, similar to the central compact source of the Cas A SNR. We suggest that CXOU J085201.4-461753 belongs to the growing family of radio-quiet compact central sources, presumably neutron stars, recently discovered in a number of SNRs.