Natural gas hydrates are a potential source of energy and may play a role in climate change and geological hazards. Most natural gas hydrate appears to be in the form of 'structure I', with methane as the trapped guest molecule, although 'structure II' hydrate has also been identified, with guest molecules such as isobutane and propane, as well as lighter hydrocarbons. A third hydrate structure, 'structure H', which is capable of trapping larger guest molecules, has been produced in the laboratory, but it has not been confirmed that it occurs in the natural environment. Here we characterize the structure, gas content and composition, and distribution of guest molecules in a complex natural hydrate sample recovered from Barkley canyon, on the northern Cascadia margin. We show that the sample contains structure H hydrate, and thus provides direct evidence for the natural occurrence of this hydrate structure. The structure H hydrate is intimately associated with structure II hydrate, and the two structures contain more than 13 different hydrocarbon guest molecules. We also demonstrate that the stability field of the complex gas hydrate lies between those of structure II and structure H hydrates, indicating that this form of hydrate is more stable than structure I and may thus potentially be found in a wider pressure-temperature regime than can methane hydrate deposits.