The center of the eclipse path (where the moon completely blocks out the sun—totality—for the longest length of time) will pass right through Fremont County/Wind River Country, fairly close to U.S. Route 26. The path of maximum totality (the center line) runs fairly close to the Old Yellowstone Highway (US Hwy. 26 and the northern portion of U.S. Hwy. 287) between Dubois and Casper. Between Pavillion and Shoshoni, the path follows Wyoming Highway 134.
From the moment the moon’s shadow first “touches” the sun to the moment it fully reveals the sun after passing in front of it, a full eclipse experience lasts for about 2 hours, 45 minutes. Dubois, Whiskey Mountain, Crowheart, Pavillion, Shoshoni, and major parts of the Wind River Indian Reservation lie along the center meridian, and those communities will experience totality (the moon hiding the sun completely) for two minutes, 30 seconds at about 11:35 a.m. Several towns, including Lander, are also within the central path but closer to the edges and will have shorter viewing times.
Find the exact time and extent of the eclipse at any location in Wind River Country using this interactive map created by NASA.