It was recently demonstrated that stable water bridges can form between two relatively large disjoint nanochannels, such as carbon nanotubes (CNTs), under an applied pressure drop. Such bridges are relevant to fabrication of nanostructured materials, drug delivery, water desalination devices, hydrogen fuel cells, dip-pen nanolithography, and several other applications. If the nanotubes are small enough, however, then one has only single-file hydrogen-bonded chains of water molecules. The distribution of water in such nanotubes manifests unusual physical properties that are attributed to the low number of hydrogen bonds (HBs) formed in the channel since, on average, each water molecule in a single-file chain forms only 1.7 HBs, almost half of the value for bulk water. Using extensive molecular dynamics simulations, we demonstrate that stable bridges can form even between two small disjoint CNTs that contain single-file chains of water. The structure, stability, and properties of such bridges and their dependence on the applied pressure drop and the length of the gap between the two CNTs are studied in detail, as is the distribution of the HBs. We demonstrate, in particular, that the efficiency of flow through the bridge is at maximum at a specific pressure difference.