What are the names of every single mining company in Australia that
mines for uranium and that also trades on the stock exchange in the
U.S.?

Australia only has three operating uranium mines, owned by BHP
Billiton, Rio Tinto and General Atomics of the United States.
http://www.freerepublic.com/focus/f-news/1608626/posts


Of the three companies, only  BHP Billiton Limited (ADR) (Public,
NYSE:BHP)  is traded on the New York Stock Exchange.
http://finance.google.com/finance?q=BHP+Billiton+Ltd&btnG=Search


Rio Tinto Limited  (Public, ASX:RIO)  is a public company traded on
the Australian Stock Exchange.
http://finance.google.com/finance?cid=676309


General Atomics is a Private Company
http://finance.google.com/finance?cid=4172339


Search criteria
Australian uranium mining companies 
Public company
NY Stock exchange


I hope this helps,  
Best regards, 
Bobbie7

Actually, the answer is partially correct with regards to Rio Tinto. 
Rio Tinto is publicly traded on the NY Stock Exchange under the symbol
NYSE:RTP (current price around $223 per share) which stands for Rio
Tinto PLC.  According to this linked Reuters profile -
http://www.investor.reuters.wallst.com/stocks/company-profile.asp?rpc=66&ticker=RTP.
- which gives additional detail on their uranium mining operations.

"Rio Tinto plc and Rio Tinto Limited (Rio Tinto) operate as a single
business organization engaged in mining. Its major products include
Iron Ore, Energy, Industrial Minerals, Aluminum, Copper, Diamonds,
Exploration and Technology."

"Rio Tinto Energy group's coal interests are in Australia and the
United States. It also includes Rossing Uranium in Namibia and Energy
Resources of Australia, which supply uranium oxide for electricity
generation."

"Energy Resources of Australia Ltd (ERA) produces uranium oxide at the
Ranger open pit mine, 260 kilometers east of Darwin in the Northern
Territory. ERA also has title to the nearby Jabiluka mineral lease,
which in 2003 was put on long term care and maintenance. Uranium oxide
from Ranger is sold to electricity utilities in Japan, Korea, Europe
and North America."