We perform molecular dynamics simulations of water confined by hydrophobic atomically-detailed infinite surfaces that mimic the structure of cristobalite. Our simulations are performed at T=300 K, densities up to 1.4 g/cc, and inter-surface separation in the range 0.6 <=d <=1.6 nm. At large separations (d >= 1nm), we find that, as the density decreases, the liquid becomes unstable transforming into a vapor phase; the liquid-to-vapor spinodal being located at ~0.7 g/cc. At intermediate separations (approximately 0.7 <=d <=0.9 nm), a liquid-liquid (LL) transition occurs before the liquid-to-vapor spinodal is observed. At smaller separations (approximately d = 0.6nm), where two water layers fit between the surfaces, the LL transition is not observed. Instead, a bilayer ice is formed before the liquid-vapor spinodal is reached. We study the structure of the liquids and bilayer ice and find that the structures of these phases are correlated with the surface structure suggesting, therefore, that the bilayer ice and the LL transition are induced by the confinement. A (lateral) pressure-d phase diagram with the vapor, crystal and liquid phases is proposed. In this phase diagram the two liquids are separated by a first-order transition line ending at two critical points, one at "large" d while the other corresponds to "small" d.