AUS Controller providing OJT chronicled a near loss of separation event when two aircraft being handed off; involving two sectors; and in the process of changing frequencies required immediate turns to avoid a conflict.

Narrative

I was working Radar West (RW) and conducting OJT. Aircraft #1 was pointed out to Radar West from Radar East (RE) and was handed off to the GRK adjacent facility that the aircraft was entering. At the same time the GRK Controller was requesting a particular altitude by a C210 (Aircraft #2) who was entering the Radar West airspace. The request was denied because of the MVA on the route of flight. The GRK Controller requested 5000 FT and our control. We accepted. I did not like the situation and told the Trainee to keep an eye on this situation with Aircraft #1 and #2. The situation was evolving and I took over the line to GRK and said; 'Reference Aircraft #1 approved.' They announced that they were not yet talking to Aircraft #1. The situation was now imminent and I yelled over to the Radar East Controller that GRK is not talking to Aircraft #2. Hurriedly; the Radar East Controller turned the aircraft to a northerly heading. At the same time I was now; for the first time; talking to aircraft #2 and was trying to get visual separation. The C210 pilot advised that he was in IMC. This action avoided an operational error and a possible midair.