An X-ray study was made to examine whether some part of the soft X-ray background is coming from hot gas in the Local Group. For this purpose, four consecutive pointings were made with ASCA toward a sky region between M 31 and M 33, which is close to the direction of the center of the Local Group. By comparing the X-ray surface brightness in this sky direction with that in another blank sky region near the north equatorial pole, an upper limit on any soft excess X-ray background was determined to be 2.8×10 −9 erg cm −2 s −1 sr −1 with a 90% confidence level statistical error. Assuming an optically thin thermal bremsstrahlung energy spectrum (Raymond-Smith model) for a temperature of 1 keV and a β-model electron density distribution for a core radius of 100 kpc for the X-ray halo, the upper limit of the central plasma density was obtained to be 1.3×10 −4 cm −3 . The plasma column density is too low to contribute significantly to the observed quadrupole anisotropy in the cosmic microwave background.