Along with a firm foam mattress, waterproof mattress cover and sheet, they come as part of a pack along with baby supplies and educational materials for new parents.
It is thought the limited width of the baby box prevents infants rolling. They are made of thick cardboard and can more easily be moved around the home than a Moses basket or cot.
It is also thought to reduce the likelihood of babies being left in adult beds or on sofas.
The baby box scheme proved a social leveller in Finland, as they were given out to all parents regardless of their economic background. Those being used in Britain are made by US social enterprise The Baby Box Co and are being donated to the Imperial College Healthcare NHS Trust for free.
If the trial is successful, the brand will continue to supply free boxes to the hospital, which sees about 125 births a week. They can be bought separately online for between £52 and £168.
The contents also include a baby thermometer, teething ring, sleepsuit, hat and booties, wash cloth, nappies, baby wipes and baby wash. All babies using the boxes will be monitored for eight months by the Trust.
Mothers will also be given access to online videos and guides featuring midwives and doctors, as well as connect with other new parents in their area. Consultant obstetrician
Dr Karen Joash, who is leading the trial, said: "For too many years the UK has fallen behind its European counterparts when it comes to reducing infant mortality.
"These boxes and the education resources that sit alongside them have been proven to help reduce the infant mortality rate in Finland and we hope that these results could be replicated in the UK."
Louise Silverton, of the Royal College of Midwives, called the scheme an "interesting idea" that could encourage parents to put babies on their backs to sleep.
Premature or low birthweight babies are most at risk and experts recommend parents reduce the risks by not smoking around their baby and by always lying them on their back when they sleep.
