As has already been stated, this would likely result in a quagmire somewhere between the Rhine and the Polish-Soviet frontier, followed by some sort of negotiated peace, and a very militarised border. That said, however, if this became a war of attrition rather than simply one of conquest/ defence, the USSR wouldn't stand a chance. It simply couldn't keep up with the US. While others would point out the vastly greater US industrial power, you must also take into account the fact that by this time, the US possessed much greater - and fresher - manpower reserves.
If this went for very long, you'd get a nuclear America with a huge industrial, economic and numerical advantage fighting against demoralised, starving - there was a famine in the USSR during 1946 in OTL, imagine how much worse it gets without lend-lease grain or the trucks to carry it in, combined with a fluid battlefield situation which always makes supply difficult - and ill-equipped Red Army. On the other hand, the USSR is impossible to occupy in the way that Germany was; too large, the population too dispersed, industry in a very inaccessible area (Siberia). So again, you'd likely end up with some sort of negotiated peace, only with whoever replaced Stalin after Moscow got nuked taking over.
@Racsoviale: This is actually a myth. While the US did only have two nukes when they went after Japan, people confuse correlation with causation in this one. They only built two nukes because they didn't need to build any more. Original plans were to construct something like ten a year or more. These plans were abandoned after Japan surrendered, with the US siphoning those resources into occupying Japan and Germany, rather than bombing either to the stone age. A continuation of the war in Europe, with the Soviets as the new enemy, would have resulted in those original plans being followed, likely with the bombs being used in Europe rather than the Pacific, seeing as how Japan was already beaten, whereas the Soviets would have enjoyed considerable initial success.