Cuban said that inopportune texts were all too common in business and private lives.
"It pops up all the time," he added.
To that end, he backed development of Cyber Dust.
"After I send you a message on Cyber Dust and you read it, it's gone forever," he said.
Cuban said that he uses the app for 80 percent of his messages, save a few text-messaging holdouts.
"It's cut down on my email, too," he said. "A lot of folks are calling it 'email 2.0' because when you email somebody, we tend to be long-worded and it just builds up, and our email boxes back up. On Cyber Dust, when I send you a message, it has to be short by definition because you don't have forever to read it. And your responses are short, and you reply immediately. And so, I've cut my email dramatically as well."
Cuban pushed back against the idea that the app was meant to hide anything.
"Look, 99.999 percent of traders don't have inside information. It's just a better way to communicate," he said. "But if I want it to be private between us, and if for some reason the SEC goes bonkers like they always do and decide to assign some sort of nefarious action to you, they can't do it."
While ephemeral-messaging app Snapchat received a new $10 billion valuation, based on a reported $20 million investment from Kleiner Perkins, Cuban said he didn't know what Cyber Dust might be worth.
Read More Kleiner Perkins $20M investment values Snapchat at nearly $10B: DJ, citing sources
Would he sell for, say, $6 billion?
"Oh, yeah," he said. "Of course I'm taking it. I'm not an idiot."