Boeing 767, CF-BEG, had departed Sydney at 1352 hours on a regular public transport flight to Auckland and was climbing to Flight Level (FL) 370. The aircraft was being controlled by Sydney Arrivals North who maintained it at FL 270 due to the presence of an opposite direction aircraft. When the controller was able to radar separate these aircraft he climbed CF-BEG to FL 290 as other arriving traffic precluded the use of a higher level at that time. Arrivals North then handed over the control of CF-BEG to Sector 5 who immediately asked how separation was being achieved with N68065, a Douglas DC10 aircraft maintaining FL280, which was inbound to Sydney from Auckland and on the reciprocal track to CF-BEG. The Arrivals controller claimed to have no knowledge of N68065 and immediate checks were made of the aircraft altitudes. This check revealed that CF-BEG was maintaining FL 290 and therefore a 1,000 ft separation standard had been achieved. The arrivals north flight progress strip had been placed on the console but the controller had not placed it in the flight progress board to indicate the confliction. The arrivals controller was performing the combined duties of Arrivals North and Arrivals South at the time due to a comfort break requirement of the other controller. Sector 5 was being performed by a controller under training supervised by a suitably rated check controller. The trainee had recognised the conflict between N68065 and CF-BEG and had twice attempted to pass a separation requirement to Arrivals North. Each time the arrivals controller was busy and the co-ordination was unsuccessful. As the sector traffic situation increased in workload due to poor weather and several aircraft diversions, the trainee forgot to make a third attempt to pass this information and the check controller was unable to arrange for it to be done. The approach procedural controller had received the co-ordination on N68065 from sector 5 and had updated the information on the flight progress strip and passed it to the arrivals north controller who did not immediately place the strip in the flight progress board. The strip was still in front of the arrivals controller at the time of the occurrence. Workload was considered to be moderate. Findings 1. Arrivals north controller was on combine and his workload was considered moderate. 2. The arrivals north controller had not placed the flight progress strip for N68065 onto the flight progress board and was thus unaware of its operation. 3. Due to the moderate workload at the arrivals north position, the controller was unable to accept incoming verbal communications which would have alerted him to the possibility of a conflict between N68065 and CF-BEG.