We present observations of magnetic fields of the very quiet Sun near disk center using the Spectro-Polarimeter of the Solar Optical Telescope aboard the Hinode satellite. These observations reveal for the first time the ubiquitous presence of horizontal magnetic fields in the internetwork regions. The horizontal fields are spatially distinct from the vertical fields, demonstrating that they are not arising mainly from buffeting of vertical flux tubes by the granular convection. The horizontal component has an average ``apparent flux density'' of 55Mxcm-2 (assuming the horizontal field structures are spatially resolved), in contrast to the average apparent vertical flux density of 11Mxcm-2. The vertical fields reside mainly in the intergranular lanes, whereas the horizontal fields occur mainly over the bright granules, with a preference to be near the outside edge of the bright granules. The large apparent imbalance of vertical and horizontal flux densities is discussed, and several scenarios are presented to explain this imbalance.