The centre in Deptford, south-east London, provides training for young people from disadvantaged backgrounds.
It was founded by Baroness Lawrence in memory of her son who was killed in a racist attack by white youths in 1993.
The Cambridges also visited XLP urban Christian charity in Gypsy Hill.
Baroness Lawrence showed the royal couple the Harris Academy law workshop that aims to raise pupils' career ambitions through lectures and mentoring.
She also showed them a digital journalism project called The Write Way that encourages young people to develop their literacy and IT skills.
"It was a quite amazing visit," she said. "The duke and duchess spent a really good time talking to our young people and showed genuine interest in what they were doing."
Stephen Lawrence was stabbed to death at the age of 18 by a group of white youths near a bus stop in Eltham, south-east London.
He aspired to become an architect and the charity was set up by his mother in his memory in 1998, originally to provide support and bursaries to budding architects.
Staff and volunteers at the centre wished the couple good luck with their new baby, which William said was due "any day now".
Catherine is believed to be around eight months pregnant and is expected to give birth at the end of April.
Before leaving the centre Stephen Lawrence's 10-year-old niece Mia presented the duchess with flowers and revealed that she liked to cook shepherd's pie, to which William replied: "I quite like shepherd's pie, I will have to pop round and try it."
William and Catherine then visited Christ Church in Gypsy Hill, south London, to find out about the work of XLP.
XLP describes itself as a Christian urban youth charity operating in seven London boroughs that helps  young people from deprived inner city estates to realise their potential.
Many of the people it seeks to help are coping with family breakdown, unemployment, gang life and educational failure.
"For these young people to know that the royal couple is interested in their lives and wants to hear what they have to say is hugely significant, as well as inspiring for them," the charity said.

The 13-year-old was cycling on the green trail near the Ranger Station at about 17:00 on Monday when he was approached by a man.
The suspect was 5ft 2in (1.57m) tall, of medium to large build, and had white hair but was balding. He was wearing a mustard coloured jacket, a red jumper, jeans and brown dress shoes.
Officers have appealed for information.
They are particularly keen to trace a dog walker who was in the area at the time. He is described as being in his 30s, and was walking a black and white bulldog-type dog.
The Duke and Duchess of Cambridge visited the Stephen Lawrence Centre and a youth charity in south London - on Catherine's last day of official events before the birth of her second child.
[NEXT_CONCEPT]
Police are investigating after a teenage boy was indecently assaulted in Dundee's Templeton Woods.