This was a pre March 16, 2013 answer. The question specifically stated that it was a late Feb. 2013 question. Now it is no longer valid
Since the AIA first-to-file is not yet in effect, B's patent is certainly under the "old" , current law. If A moves quickly he can get in an application by March 15th under the old law also. That application could be made by editing the content of B's application to the extent needed to only include inventions actually made by A. Under 35 U.S.C. 135 Interferences, the applicant A can suggest an interference with B's issued patent as long as (as per 135(b)) A's application is filed within a year of B's publication or patenting.
This timing situation could be the case if B's patent was issued less than a year ago and B's application had a non-publication request or if B's application went through the system much more quickly than usual - under a prioritized or accelerated examination, for example. In the normal course of events, if B's patent recently issued, it is likely to have been published longer than a year ago. That would prevent an interference.
In the interference proceeding A and B would provide evidence of date of conception and diligence to reduction-to-practice and A may be able to prevail and get the patent or A and B might reach a settlement.
Of course this system will be a thing of the past in several years when no pre f-2-f applications are in the pipeline. Under the new scheme, interference is replaced by derivation proceedings. Proving you were first will not be the direct issue but rather proving the other person got it from you rather than inventing it themselves. Does not apply to the facts of the hypothetical question since it refers to a recently issued patent as of late Feb. 2013.