Lawyers say the admission by bosses at Good Hope Hospital in Sutton Coldfield paves the way for a lifetime care and rehabilitation package to be agreed.
Cassandra Wiggin, known as Cassie, has severe cerebral palsy and is completely dependent on 24-hour care because midwives did not monitor her heart rate as she was being delivered, causing it to drop dangerously low and her brain to become starved of oxygen.
Her parents, Sarah Myatt-Maley and James Wiggin, instructed lawyers to investigate whether more could have been done to help her and to help them get support and access to specialist rehabilitation services.
Lawyers from the firm Irwin Mitchell say Heart of England NHS Foundation Trust failed to admit Cassie's mother on the labour ward in good time, did not induce her labour properly and did not monitor the unborn baby properly as Cassie's heart rate was normal.
It was also said that had Cassie been delivered by late afternoon to early evening on March 13 by emergency C-section, instead of 11.30pm, the brain damage would, in all likelihood, have been avoided.
Her parents today said they are relieved a settlement can now be negotiated but added it remains difficult not to be angry as Cassie is wheelchair dependant, unable to talk and feed herself and they feel this could have been avoided.
Sarah, aged 24, said: "Our little girl had the worst start to life as a result of the hospital's mistakes and that has been very difficult for us as a family to come to terms with.
"She has been diagnosed with severe cerebral palsy and has severe learning difficulties, epilepsy and visual impairment. She cannot walk, talk or communicate in any way with us or her little sister. She is now nil by mouth and can only be fed through her gastrostomy tube. Cassie is regularly in and out of hospital with chest infections and will be entirely reliant on us for the rest of her life.
"We are pleased and relived that the trust have accepted responsibility for the mistakes made when Cassie was born and now this paves the way for us to be to continue to provide her with the very best care for the rest of her life so she can grow and develop."
On March 9, 2010, following a normal pregnancy, Sarah was admitted to Good Hope. She was 11 days overdue so she was advised she would need to be induced.
Over the next few days medical staff monitored Sarah's progress, but she was not induced. On March 13, which was 15 days past her due date, the couple say a nurse attached a heart monitor at 8.38am and a procedure was performed to make her waters to break. She was then told that arrangements would be made to transfer her to the delivery suite to have her baby.
At 11pm she was seen by a nurse again, and a heart monitor was attached and was taken to the operating theatre for an emergency caesarean section and delivered Cassie.
When Cassie was born she was not breathing and needed resuscitation and was transferred to New Cross Hospital in Wolverhampton for extensive brain cooling as she had been starved of oxygen. She spent seven days in intensive care and a further eight days in the high dependency unit before she was allowed home with her parents.
Hospital chiefs have apologised and said action has been taken to improve the safety of patients.
In a statement the trust said: "The trust admitted liability at the earliest opportunity, following investigations into the events of Cassie's birth. An apology letter has been sent and this includes information about lessons learned and steps implemented to improve patient safety. Our respective legal teams will now be working towards determining the appropriate level of compensation to ensure Cassie's lifelong needs are met."
Emma Rush, a specialist medical lawyer at Irwin Mitchell's Birmingham office representing the family, said: "It is a huge relief for Cassie's family that the trust has admitted liability for the brain damage she suffered as it means we can now negotiate a lifetime care package that reflects her complex care needs.
"Cassie requires 24-hour care and whilst her family have shown her complete dedication, she needs the support of one-to-one care and specialist rehabilitation services to ensure she thrives to her full potential.
"Unfortunately the mistakes made during Cassie's birth are not uncommon and trusts across the NHS need to ensure that they are improving training and resource in maternity services so that mums and their babies are given the best and safest care possible.
"We would welcome a letter of apology sent from the trust to Cassie and her parents, however we believe it also needs to confirm exactly what steps have been taken to learn lessons and improve standards form the errors made to protect future patient safety."