Sergeant Steve Watt, of Queenstown, said two men who were roped together on a route called The Grand Traverse fell to their deaths around midday.
One body was recovered this afternoon and efforts were ongoing into the evening to recover the second body.
However, the operation was suspended and is due to resume at 6am tomorrow.
The pair were on a guided climb with Wanaka-based Aspiring Guides.
Its general manager Jess Marriott said the company was "doing everything we can" to support the police investigation in to the incident.
The company said it could not give further details as police were investigating.
WorkSafe is also making initial inquiries.
A police spokeswoman said police were alerted that two climbers had fallen from the side of a cliff face about 11.45am.
The climbers were near Lake Alta, which is near Double Cone on the mountain range.
Aspiring Guides' website said the Grand Traverse was suitable for people with hiking experience and "a head for heights". It also offered alpine rock climbing direct to the summit of either Single or Double Cone.
Part of the trek required scrambling along an exposed rocky ridge, with a near-vertical 2km view down to Queenstown and Lake Wakatipu, the website said.
Queenstown Mountain Guides' website said the trek usually took eight to 10 hours to complete.
- additional reporting by Guy Williams and NZ Herald
