The Australian Antarctic Division (AAD) said the incident happened yesterday when the ship was on a resupply mission to the Mawson Station, with gales of more than 130 kilometres per hour whipping the base.
"A total of 68 expeditioners and crew are on board, and all remain safe and well," it said, adding that the icebreaker ran aground at West Arm in Horseshoe Harbour.
Australian Antarctic Division director Nick Gales said they would only be transferred the short distance to Mawson when conditions improved.
The blizzard conditions have been hindering a full assessment of the ship but the division said there was a breach in the hull "into a space that is usually flooded with ballast water".
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"This breach has occurred in an area of the ship that poses no risk to the stability of the vessel or of fuel leaking into the environment," it said.
"Attempts to refloat the vessel will occur when the weather conditions ease. It will take a minimum of three days for the ship's crew to complete a full assessment of the ship once the vessel is afloat again."
The AAD said the US Antarctic programme had offered assistance and was flying in a LC130 aircraft to get them to the Casey Station where a bigger plane can land in the coming days to transport them out.
Gales said various contingencies were also being looked at for those on the Aurora.
"We're in regular touch with the operators of the other national programs so we're looking at ship and aviation options. It's too early to say whether we need those options and secondly which of those will prove the most effective to get people home quickly," he said.