Polyurethanes are made in a one-shot process from one or more polyols having a hydroxyl equivalent weight of at least 350, wherein at least 50% of the weight of iii) is a hydroxyl-containing polymer of propylene oxide, the hydroxyl-containing polymer of propylene oxide having a hydroxyl equivalent weight of at least 350, an average of 1.8 to 3 hydroxyl groups per molecule of which hydroxyl groups 40 to 70% are primary hydroxyl groups, an oxyethylene content of no greater than 10% by weight based on the weight of the polymer and a polydispersity of 1.175 or less. The polyurethanes exhibit excellent mechanical properties, are highly hygroscopic and cured rapidly.