Joanna Cohen, New York Times

If any parent doubts the power of the Disney animated film "Frozen," perhaps they should talk to Willie Geist, a "Today" show host.

The day Geist left for Sochi, Russia, to cover the Winter Olympics for three weeks, his daughter, 6, and son, 4, were distraught, sobbing and clinging to his legs, he said. Moments after he pulled out of his driveway, his wife texted him a picture of their children jumping with excitement.

"She had just told them they were going to see ‘Frozen,’" Geist said. "That joy far, far outweighed the sorrow of their father leaving for a faraway land for nearly a month. Completely erased the tears."

Since opening in November, "Frozen" has taken in $1.19 billion worldwide, and the movie’s soundtrack has sold 2.7 million copies in the United States alone.

It may be surprising that a Disney movie centered around two female characters, the sisters Elsa and Anna, could drive such staggering sales. But the real eye-opener is the strength and scope of the film’s grip on children, girls and boys alike. Parents are using words like "obsessed" and "cultlike" to describe the force with which the film is capturing little hearts and minds unlike any in recent memory.

"We have been doing so many ‘Frozen’ cakes, it’s crazy," said Rachel Thebault, owner of Tribeca Treats, a bakery in downtown Manhattan whose clients have included Jessica Biel and the New York Knicks star Carmelo Anthony. "In the seven years since we’ve been open, I have seriously not seen any other trend come close to this."

With stores across the country sold out of the most coveted merchandise, desperate parents are turning to eBay, where, according to Bloomberg News, an original Elsa dress was recently offered for $1,600.

While brawling over the last must-have toy on the shelf is hardly unprecedented, "Frozen" is taking the madness to another level. A Manhattan mother said her son insists she call him Elsa while he runs around the house pretending to freeze everything. Another mother lamented that while her guests were around the table at Passover, her preschooler was in the other room singing "Let It Go" at the top of his lungs.

Idina Menzel’s rendition of what has become the movie’s anthem, "Let It Go," is a phenomenon unto itself, producing a level of passion and exuberance in children the likes of which some parents have never seen.

The song jump-starts Jack Solo­mon, 4, who is often slow to rise and grumpy in the morning. One day his father, Steven, entered his room playing "Let It Go" on the computer. Jack leapt out of bed, beaming and ready to start the day.

"It’s like a drug," said Jack’s mother, Moira Jaffe-Solomon. "I only use it on days we need to be somewhere and he’s still sleeping."

In addition to the global appeal of its soundtrack, the film’s extraordinary success may also be explained by the balance it strikes between classic story elements and modern twists.

"‘Frozen’ both honors and subverts our expectations of a princess movie," said the screenwriter John August. "There’s a handsome prince but he’s the villain. True love is the answer, but familial love, not romantic. Tropes have their place but ‘Frozen’ manages to refocus them."

Danielle Human of Chicago finds Anna’s loyalty, bravery and determination empowering for her 6-year-old daughter, Ruby.

"It makes my heart warm that she can finally connect with a girl who is lovely and witty and funny and real," Human said.

Not all parents see "Frozen" as having such a positive influence. Kate Schaper of Manhattan said her son, Benjamin, 4, throws lines from the movie at her in dramatic fashion like, "I can’t live like this anymore."

Leslie Talmadge Ellwood of Manhattan has seen the movie make a different yet equally significant impression on her children, Natalie, 4, and Ben, 10.

"Natalie loves ‘Let It Go,’ but she has no idea that the lyrics are about dropping a facade," she said. "It’s a little jarring to hear such a young girl sing lines like, ‘Conceal, don’t feel.’"

But Talmadge Ellwood has been pleasantly surprised that "Frozen" has paved the way into several meaningful conversations with her son. He was most struck by the handsome and gallant Hans, who (spoiler alert) pretends to be in love with Anna when he is really after the throne.

"We’ve talked a lot about that, how people can act a certain way because they want something from you," Talmadge Ellwood said. "I’ve told Ben never to act that way. It’s insincere and hurtful."

Joan Rosen, who has taught pre-kindergarten at the Ethical Culture school in Manhattan for 42 years, said the impact of "Frozen" on her students has been intense.

Themes related to "Frozen" have become a regular part of the play in her classroom with unexpected results. Girls who don’t usually build in the block area are now doing so, making "ice castles." The "Frozen" fervor has also proved distracting at times, so Rosen has instituted some rules. The children are allowed to sing one song from the movie three days a week, right before lunch, or any time during outdoor play.

"‘Frozen’ is visually very beautiful, and the music is so appealing, even to me," Rosen said. "But speaking as an adult woman, I’m pretty much over it."

She is not alone. Scott S. Kramer, a father of two in Los Angeles, tapped into the growing sense of "Frozen" fatigue by writing a "Let It Go" parody expressing the frustrations of those who had reached their breaking point. The video has gotten nearly 1.5 million views on YouTube.

The insatiable appetite for all things "Frozen" has parents like Geist a little nervous.

"I’m concerned they don’t have a sequel ready yet," he said. "I can only hold off these kids for so long."