Carefully dated? At that claimed age its more like hyper-assumed. What sceintific method has confirmed the beginning conditions were known; the beginning ratio of daughter to parent Isotope were known (zero date problem)l that there was a constant decay rate; that there was no leaching or addition of parent or daughter isotopes; that the forgoing assumptions have been valid for billions of years, etc. Not to mention the extreme difficulty in measuring precisely very small amounts of the various isotopes.
http://www.detectingdesign.com/radiometricdating.html
The article clearly identifies the source of this exceptionally well-preserved fossil (NOT soft tissue) as Hamilton Quarry in Kansas.
Hamilton Quarry is a well-known site of many well-preserved fossils from the Carboniferous era.
World-wide Carboniferous deposits are dated from around 360 million years ago to about 300 million years ago, and account for much of the world's coal supplies.
How is geological dating accomplished?
The gold-standard is radiometric dating, for which there are dozens of different materials that can be used, depending on a stratum's age.
For just two examples: carbon-14 is very accurate back to around 60,000 years ago, while Uranium-lead is accurate within around +-2 million years, but all the way back to billions of years ago.
Uranium-lead samples also come with built-in cross-checks using both U-235 and U-238.
Are all of these methods fool-proof?
No, of course not, they all have to be double-checked, cross-checked and re-checked to be sure the answers are consistent with everything else known about the samples.
But given that many thousands of such tests have been performed, over many decades and all over the globe, the consistencies of results are about as good as you can ever get in natural-science, and support every other theory we have (i.e., evolution) about the natural history of ancient Earth.