We present a high resolution (R~500) X-ray spectrum of the bright quasar H1821+643 (z=0.297), obtained in a 470 ksec Chandra observation. We search for X-ray absorption by highly ionized metal species, OVII and OVIII in particular, at the redshifts of the six intervening OVI absorption systems known from UV studies. We detect features with >~2-sigma significance at the predicted OVII and OVIII wavelengths of one OVI system, at the OVII wavelength of a second, and at the NeIX wavelength of a third. We find two additional features of comparable strength (one OVII and one OVIII) within 1000 km/s of OVI redshifts. The 1-sigma constraints in the two detected OVI systems imply gas overdensities lower than the values delta>100 expected in virialized systems, suggesting that the absorption arises in lower density, filamentary structures. At the 2-sigma level, however, the physical constraints are weak. If we treat our 2-sigma detections of known OVI systems as real, but assume minimal OVII and OVIII in the other systems, we estimate [f(OVI)+f(OVII)+f(OVIII)]/f(OVI) = 32 +/- 9 for the average ratio of all highly ionized oxygen species to OVI. Combined with estimates of the total column density of OVI absorption per unit redshift, this ratio implies a total baryon fraction associated with detected OVI absorbers Omega_b(OVI)~0.03/h_70, a substantial fraction of the baryon density predicted by BBN, and larger than that in known low redshift components. Because of the limited S/N of the detections, these results must be treated with caution. Nonetheless, the combination of the OVI data with these X-ray forest measurements provides the most direct evidence to date for the pervasive, moderate density, shock-heated intergalactic medium predicted by leading cosmological scenarios. (Abridged.)
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