Khloé Kardashian is loving motherhood. The 33-year-old reality star gave birth to her first child, a daughter named True, on April 12th.

A source close to Khloé revealed to US Magazine, "Khloé is absolutely obsessed with True, and is up with the baby in the middle of the night whenever she cries. Khloé does have a nanny, but she is hardly working because Khloé is so hands-on.”

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Khloé has spoken publicly about her struggle with infertility and her desire to become a mother. On The Ellen DeGeneres Show she explained, “I’ve wanted to have a baby for so long it’s just the perfect timing. God always has a plan and I just wanted to do it on my time and when I felt comfortable to do so.”

The Kardashians are known for their unique name choices, so fans were less than surprised when Khloé named her baby True. But there is a real history and thoughtfulness to this name choice, as Kris Jenner, matriarch of the Kardashian clan, explained via Instagram.

She said, "I’m so excited to welcome my precious little granddaughter True!!! FUN FACT... my Grandfather’s name on my Dad’s side was True Otis Houghton....my Dad’s name was Robert True Houghton...so i am so excited Khloe named her daughter True!!!"

Baby True was born in the midst of allegations of infidelity revolving her father, NBA star Tristan Thompson. Photos and videos were released that showed what appeared to be Thompson engaged with questionable behavior with at least three other women, throughout Khloé's pregnancy.

Sources place Thompson at the hospital at the time of True's delivery, but also affirm that he is not living in his Cleveland house with Khloé at this time. Yesterday Thompson joined his team, the Cavaliers, in Indianapolis for game 3 of NBA playoffs. This is his first time away since Khloé gave birth a little over a week ago.

It is unclear at this time the status of the celebrity couple's relationship, but a source told People earlier in the week, “Khloé really wants to be able to forgive and trust Tristan again and have the life she hoped to have with him. She doesn’t know if she’ll be able to do that, but that’s the goal. She loves the life they had, she loves the baby they have together, and she still loves the dream of being a family together. If she can find a way to rebuild with Tristan, she will.”