After repeated delays due to poor visibility, and no sign of the situation improving, officials gave up on attempts to complete the race in one day.
"The plan now is to do the second run tomorrow," race director Atle Skaardal said, with the new start set for 09:30 local (17:30 GMT) from the regular hut.
"We knew it would be a very difficult day today. We know also it will be a similar difficult day tomorrow," said Skaardal.
Asked what would happen if the run could not be held on Thursday, the Norwegian said it would have to be discussed within the Games' emergency group. However, he said the race could not be valid without a second run.
Slaloms are scheduled for Friday and Saturday, although the men use a different piste. Skaardal said he was flying home on Sunday.
"We were quite lucky and were able to do a good and fair first run and due to the weather change were not able to pull off a second run," he continued.
"The jury were unanimous in the decision and nobody felt it would be fair to cancel the first run."
Elisabeth Goergl, well placed to become the first Austrian to win the women's giant slalom since it was introduced in 1952, led France's Taina Barioz by 0.02 seconds after the first run, with two more Austrians close behind.
"I think it is a fair decision from the jury to do it tomorrow because we don't want a lottery at such an important race like this," Goergl said. "Let's hope that we have fair conditions tomorrow and a good race."
The regulations state only that "whenever possible" both runs should be completed on the same day.
The decision to proceed with only the second run, rather than a completely new race, ended any hopes of American Lindsey Vonn being able to compete after she crashed out of the first leg.
And Julia Mancuso's hopes of retaining her Olympic title were wrecked by team-mate Vonn crashing just before her first run.
Mancuso was given the yellow flag before she reached an area of the course where Vonn had crashed out and had to be taken back up to the start with the later runners.
After her run, Mancuso broke down in tears at the side of the track.
"That was possibly the worst possible thing that could happen in the Olympics," she said.
"To get flagged on the run of your defending gold medal run that’s not something anyone ever wants and it is probably the most unexpected thing eve.
"There were tears of frustration, I knew that the conditions had gotten slower, you look at the runners past bib 20 and there is no-one punching in there. You just feel it, there is nothing coming off the snow, for me I knew that doing two runs in a row is not easy," she said.
"Going twice is not only physically exhausting but it is also very emotionally exhausting and stressful to have to go up and do it again," she said.
Having claimed two silver medals already, Mancuso was in with a chance of a third podium in her favourite event but finished 18th after going back to start later with the snow softer from rising temperatures.
"There was really nothing I could do - just go back and try to ski a good race and to do everything that was in my control, I thought I did a really clean run and skied well.
Mancuso, who spent the rest of the afternoon waiting for the second run by listening to music and chatting with her British friend Chemmy Alcott, said she was disappointed with how the incident had been allowed to develop.
"Just the whole situation could have been dealt with somewhat better, the fact that they didn’t flag me (earlier) or just not start me.
"It just seems like in my mind there are just so many more scenarios that could have happened.
"When someone has a crash, you should stop the course. If someone skis out on a GS course then okay you run the next racer but not if there is a crash," she said.
Skaardal said that procedure had been followed correctly and the jury member where Vonn crashed had called in promptly.
"He confirmed it’s not safe to let her run by and we needed to stop her," said Skaardal.
Alcott clocked a time of 1:17.53 in her opening run, putting her 2.41 seconds off the pace in 25th.
The US team's nightmare left Goergl, one of three Austrians in the top four, as favourite to add a gold to her downhill bronze.
She was 0.02 quicker than Barioz with Austrian team-mate Kathrin Zettel in third place and with 0.16 to make up. Another Austrian, Eva-Maria Brem, was fourth.
"I really like these conditions," said Goergl as the snow fell steadily from leaden skies. "I haven't felt too confident in giant slalom this season but I'm really focused on this race."
Barioz, in her first Olympics, was surprised with her time.
"The conditions are not too bad, even if there is a visibility problem in the middle section," she told reporters. "The course is well prepared and the conditions are the same for everybody."