The Sindarin name for the eighth month of the year, used in the calendars of the Dúnedain, and equivalent to QuenyaÚrimë, or approximately to modern August. The name comes from the Elvish root word ur-, 'heat', referring to the long hot days common in this month.
In the original King's Reckoning of Númenor, Úrui had run across a period that would be equivalent to that from 24 July through to 22 August on a modern calendar. When Mardil revised the calendar to create the Stewards' Reckoning, he made some changes that affected Úrui. Most directly, the length of the preceding month of Cermië was reduced from thirty-one days to a standard length of thirty. So, by this Stewards' Reckoning, Úrui now began and ended a day earlier (running from modern 23 July to 21 August). Among the Hobbits, this month was known as Wedmath (and its dating on the Shire Calendar followed the old King's Reckoning rather than the modified Stewards' Reckoning).
There are few historical events recorded as occurring during this month, but one important exception was in the year III 2510. It was in Úrui of that year that Cirion and Eorl met in a hallow on the slopes of Amon Anwar, the Halifirien, and swore the oaths that brought the land of Rohan into existence.
Centuries later, in the same summertime period of III 3019, the War of the Ring had been over for many months. In this year, Úrui saw the funeral train of Théoden reach Edoras, where the old King was entombed on the Barrowfield. Afterward, the Travellers and their companions journeyed on northward, and near the end of the month they encountered Saruman as they passed through the wilds of Enedwaith.
Later editions of The Lord of the Rings spell the name Úrui (and its Quenya equivalent Úrimë) with an acute accent over the initial 'U'. This accent was omitted in earlier editions of the book, where the names are spelt Urui and Urimë, without initial accents.