Martha Stewart is a grandma

(CBS) Alexis Stewart, Martha Stewart's only child, has welcomed a daughter named Jude People magaine reports.

The baby girl was born via gestational carrier on Tuesday.

On her radio show, "Whatever with Alexis & Jennifer," Alexis seemed to lay down the law about how her mother will be known to Jude.

"Martha is going to be called, 'Martha,'" Alexis said. "She seemed to have no problem with it. ... I was like, 'Do you really want to be 'Grandma'? I don't like the term 'Grandma'!"

Pictures: Martha Stewart gets out
Pictures: Martha Stewart

People reported Alexis, 45, has been open about her struggle to conceive a child. She told the publication in 2007 she was spending more than $20,000 each month on fertility treatments.

She said at the time, "Having my own kid is the most important thing in my life. I'm trying everything I can."

Pictures: Martha gets an apprentice

And Alexis isn't the only one who's basking in the new addition to the family. Alexis said her mother is already spoiling Jude, bringing the baby embroidered clothes from high-end retailer Bergdorf Goodman.

Speaking of the gift, Alexis said, "I'm like, 'Wow, I guess I'm going to be allowed to buy myself something pretty nice, because all this stuff's going back!' If the kid can't see it, why the 100 dollar (clothes)? Only books, child-appropriate or whatever: that's the only thing that I want. No toys please!"