Can cord blood cure leukemia?

Yes, stem cell transplants with cord blood have been used to cure both children and adults with leukemia since the early 1990's. To date, there have been over 35,000 cord blood transplants world-wide, and most of them were for leukemias and other blood disorders (Ballen Verter Kurtzberg 2015). A study published in the New England Journal of Medicine (NEJM) in Sept 2016 compared cord blood transplants versus bone marrow transplants for leukemia patients. The two groups had comparable survival post-transplant, but the cord blood patients tended to live longer and most importantly the cord blood patients were less likely to relapse.

The important caveat is that children with leukemia or another blood disorder must receive a cord blood transplant from a donor, NOT their own cord blood. It turns out that when children and even adolescents develop leukemia, they were born with the genetic defect that triggered the leukemia... hence it is not safe to give them a transplant with their own cord blood because it probably carries the mutation for leukemia.

References:
Backtracking leukemia to birth: Gale KB et al. 1997; Proc Natl Acad Sci USA. 94(25):13950-4. PMID:9391133
Backtracking leukemia to birth: Janet D. Rowley 1998; Nature Medicine 4:150-1 PMID:9461182
Ballen KK, Verter F, Kurtzberg J 2015; Bone Marrow Transplantation 50(10):1271-8. doi:10.1038/bmt.2015.124
HealthDay article describing study in Sept 2016 NEJM: Cord Blood Transplants Show Promise in Leukemia Treatment
Filippo Milano, et al. 2016; NEJM 375:944-953. DOI:10.1056/NEJMoa1602074