Air carrier Captain reported an engine malfunction in climb nearing cruise altitude. The Captain diverted the flight to an alternate airport; took over the controls for an overweight landing; and taxied to a gate.

Narrative

On climb almost to our cruise altitude the number one engine oil quantity dropped to zero. We did the QRH checklist for the oil quantity then I had the FO fly while I called dispatch and maintenance control. While discussing options with maintenance and dispatch the oil pressure for engine number one started to fall; going amber then red. I ended the call so with dispatch telling them we would be diverting to ZZZ. With the FO continuing to fly we performed the QRH procedure for the engine oil pressure which led us to do a precautionary engine shut down and diversion. We [requested priority handling] and diverted to ZZZ. Due to our weight of the aircraft we had to preform an over weight landing as well. I took the aircraft back from the FO on final approach and preformed the landing. After landing we were checked out by ARFF (Airport Rescue and Firefighting) then once it was deemed safe we taxied to a gate in ZZZ. Cause: The oil levels were all good during preflight. Everyone worked great together to make this event go as smoothly as possible.