An article in the Los Angeles Times gives a date in August...
The tree, called Prometheus, took root at the dawn of the Bronze Age, centuries before the ancient Egyptians began construction on the pyramids at Giza. It outlasted the rise and fall of the Roman Empire, European colonialism, the Mexican-American War and the creation of the atom bomb. But it didn’t survive the chainsaw that felled it on Aug. 7, 1964, at the request of a scientist who wanted to study the tree’s rings.
Great Read: The ghost of Prometheus: a long-gone tree and the artist who resurrected its memory
However, another article gives the date as a day earlier...
It made no difference. On Aug. 6, 1964, the tree was on the ground in big sections. Cox participated in the hard work, as did two other Forest Service employees. One big rough-cut section, a foot thick and six feet long, was hauled out that same day and taken to the Nevada Northern Railway in Ely, which had the only saw big enough to slice the tough old monster into manageable sections.
Staying Alive / High in California's White Mountains grows the oldest living creature ever found
A third source mentions that, after permission was granted on Aug. 3, 1964, the intent was to cut the tree down on the 6th but reluctance on the part of a Forest Service worker to do so meant that it was only actually done on the 7th.
One Forest Service sawyer refused to cut the tree on August 6, 1964. The next day, Cox and several Forest Service crew members took turns at the saw.
Oldest Living Tree Tells All