intermittent start problem on 2002 cooper

[srdtvr]

My daughters 2002 cooper has a starting problem, turn the key and dash lights come on but engine doesn't turn - completly dead - turn key off and try agian and it starts as normal. I changed the battery in key fob as I think it could be immobiliser related- thought we'd fixed it but fault has now returned.
I still think it could be immobiliser related as it runs fine otherwise.
Any ideas anyone?
Cheers
Steve