A300 pilot reported engine oil quantity in Engine 1 decreased then later indicated zero during cruise. The flight crew diverted and after landing; shut down the affected engine.

Narrative

At FL340 just southwest of ZZZZZ; we received an Engine ADV flashing white message on ECAM systems page. Oil quantity for the #1 Engine was flashing and was showing 10 quarts. 12 is minimum for engine start. I asked the FO (First Officer) to please reference our QRH Advisory section. QRH advised if all other parameters were normal - they were - to continue normal operations.Approximately 15 minutes went by and the quantity slowly went down further until at about 30 minutes after the initial advisory it went to 0. While this was going on the FO had sent status reports to Dispatch/Maintenance Control. We were told that the A300 Fleet Manager advised to continue as well as long as all other engine parameters were normal. Dispatch notified us that they had consulted with Pratt & Whitney who conveyed that it may be an indication issue.As we were in the middle of the ocean with not many divert options readily at hand; we asked for more direct routing toward ZZZ1 from ZZZ Center. Fuel assessment was completed and deemed acceptable. We set up radios and MCDU (Multipurpose Control Display Unit) for possible divert into ZZZ2 if necessary. Once ZZZ2 was reached and engine parameters remained normal we elected to keep flying north along the coast toward ZZZ. Dispatch relayed that Maintenance Control had pinged our aircraft and they were showing that we had 1 quart of oil in the #1 Engine. There were now many divert options if we needed. As we flew north; the FO kept checking weather and landing runways at each airport that was a possibility of diverting to. Once past ZZZ3 we focused on ZZZ as our landing airport.On descent into the ZZZ area; when the autothrottles brought back the throttles to idle descent; we received a Master Warning ENG LO Press ECAM. I disconnected autothrottles and brought power back up on the #1 Engine. All parameters; except quantity; returned to operational limits. At this point we asked for priority handling with ZZZ Approach and also asked for the longer runway of XXR. I elected to conduct a flaps 15/20 approach in case we did finally end up having to shut the engine down. Approach was uneventful as long as I kept power up on the #1 Engine. After landing; we cleared at [Taxiway] 1 and proceeded south on [Taxiway] 2 to enter Ramp. Almost immediately after clearing runway we received another Master Warning ENG LO OIL Press ECAM with a conditional IF statement that stated if the oil pressure was below 70 PSI to shut it down. It showed approximately 60 PSI. We complied with the ECAM and shut down the #1 Engine and taxied to parking.