Mandy Moore Marks One Week with 'Dreamboat' Baby Ozzie, Says She's 'Exhausted' but 'So Grateful'

mandy moore baby
mandy moore baby

Stefanie Keenan/Getty Images for Olive & June, Mandy Moore/instagram

Mandy Moore is basking in newborn baby bliss.

The actress, 38, marked one week with baby boy Ozzie Thursday, sharing a heartfelt message alongside an adorable picture of the infant on Instagram.

In the sweet picture, baby Ozzie, whom the This Is Us actress and husband Taylor Goldsmith welcomed last week, rests under a blanket with his little hands behind his head.

"One week with this dreamboat. You've absolutely changed the game, Ozzie," wrote Moore. "Yes, I'm exhausted and covered in spit up and feeding you nonstop but I'm so grateful and soaking in every second of this time."

Along with Ozzie, Moore and Goldsmith share 20-month-old son Gus.

Never miss a story — sign up for PEOPLE's free daily newsletter to stay up-to-date on the best of what PEOPLE has to offer, from juicy celebrity news to compelling human interest stories.

RELATED: Mandy Moore Shares Adorable Photo of Son Gus 'Relishing His Role as Big Bro' to Baby Ozzie

On Wednesday, Moore shared a heartwarming picture of Gus holding his new baby brother in his lap, beaming with a huge smile as he looked lovingly at the infant.

Moore and Goldsmith smiled alongside Gus as they helped the toddler hold little Ozzie, who was swaddled comfortably in a blanket.

"Gus relishing his role as Big Bro 😎," Moore captioned the adorable sibling photo on Instagram.

"Ozzie is here!" wrote Moore in the sweet birth announcement last Friday, which featured a carousel of black-and-white photos from the hospital.

"Oscar Bennett Goldsmith arrived a little late but with much aplomb (and an easier/speedier delivery than his big brother, much to the delight of his parents)," she continued in the caption. "Every adage is true: our hearts have doubled in size and the immediacy of the love is astounding. He is beyond words and we are so grateful for our family of four! 💓💓💓💓."

While Gus is already proving to be a great big brother, Moore told PEOPLE last month that the toddler was "entirely unaware" he was about to get a sibling.

"I say baby brother all the time and point to my stomach, and he'll say, 'Baby brother,' but he has no idea what's going on," she said with a laugh. "We'll figure out how to best introduce the two of them, but yeah, he has no idea what's about to hit him."