Answer:
The Fort is no longer standing, but it was originally built in 1849.
From Wikipedia:
In January 1849 Worth proposed a line of ten forts to mark the Western Texas frontier from Eagle Pass to the confluence of the West Fork and Clear Fork of the Trinity River. One month later Worth died from cholera.
On June 6, 1849, Arnold established a camp on the bank of the Trinity River and named the post Camp Worth in honor of General Worth.
In August 1849 Arnold moved the camp to the North-facing bluff which overlooked the mouth of the Clear Fork of the Trinity River. The U.S. War Department officially named the post Fort Worth on November 14, 1849.
The fort was flooded the first year and moved to the top of the bluff where the courthouse sits today. No trace of the original fort remains.