We present the first experimental vibrational spectroscopy study providing direct evidence of a water phase inside single-walled carbon nanotubes that exhibits an unusual form of hydrogen-bonding due to confinement. Water adopts a stacked-ring structure inside nanotubes, forming intra- and inter-ring hydrogen bonds. The intra-ring hydrogen bonds are bulk-like while the inter-ring hydrogen bonds are relatively weak, having a distorted geometry that gives rise to a distinct OH stretching mode. The experimentally observed infrared mode at 3507 cm(-1) is assigned to vibrations of the inter-ring OH-groups based on detailed atomic-level modeling. The direct observation of unusual hydrogen bonding in nanotubes has potential implications for water in other highly confined systems, such as biological channels and nanoporous media.