The box aims to help give children the best start in life, and it contains items such as clothing, a toy, a play mat, books, a towel, a sling carrier and thermometers.
The box itself can be used as a sleeping space for newborns.
New figures show that as of December 7 this year, a box had been delivered to 68,513 newborns - the equivalent rate of around 900 a week.
This is a dip on the level recorded at the first anniversary of the box being introduced.
On August 15, 2018, a year after the policy was brought in, figures showed 52,065 boxes had been handed out, the equivalent of 1,000 per week and representing an 85% uptake by parents.
The projected cost of the initiative was £8 million for 2017/18, rising to £8.8 million for each of the following three years.
Critics at the start of the scheme suggested that giving the box to any eligible parents, including those who can easily afford the contents, did not represent the best use of public funds.
However, ministers stress feedback shows parents appreciate the universal aspect of the policy and believe it removes stigma.
SNP MSP Linda Fabiani said: “It is wonderful to see so many families across Scotland benefiting from the baby box.
“The boxes are a really positive Scottish Government initiative providing parents with many of the essentials they need to care for their child.
“From thermometers to towels, changing mats to clothes - the boxes are a fantastic way of giving children the very best start in life.
“Well over 68,000 boxes have already been provided to parents across Scotland, showing how popular they are proving with families in every corner of the country - and given how beneficial they are to parents and newborns, long may that continue.”