That all depends on exactly what the bill says.

It could make prior art irrelevant (huge step backwards, but I think unlikely). Alternatively, it could leave prior art intact. In which case, if party B patents something party A invented a short time before, and someone can prove this, then nobody gets the patent. Party A's invention is prior art to party B's patent.

Under the current system, party A would get the patent.

edit: Tracked down the bill text. I am not a lawyer; this is not legal advice; read it yourself and make your own opinions: http://www.opencongress.org/bill/112-h1249/text

  (a) Novelty; Prior Art- A person shall be entitled to a 
  patent unless--

      ‘(1) the claimed invention was patented, described in 
  a printed publication, or in public use, on sale, or 
  otherwise available to the public before the effective 
  filing date of the claimed invention; or
  
      ‘(2) the claimed invention was described in a patent 
  issued under section 151, or in an application for patent 
  published or deemed published under section 122(b), in 
  which the patent or application, as the case may be, 
  names another inventor and was effectively filed before 
  the effective filing date of the claimed invention.

I'm still trying to track down the original patent act to compare with, but it looks like prior art continues to be valid in the form of a publication or product. The dubiously-valid trick of mailing a description of the idea to yourself would not seem to work, and it's questionable whether the Internet would count as "printed", but it shouldn't be too difficult to set up a "protopatent publishing journal" that exists purely to protect inventors from having their ideas patented later.