ESA and pregnancy

Hi

I am leaving work in a few weeks due to redundancy and unable to work any more due to a medical condition and pregnancy, so will want to claim ESA. I have heard that they don't let you go on ESA after a certain stage in your pregnancy is this true? I don't need to claim maternity allowance as my employer is paying my SMP as a lump sum but I don't want to use this until closer to when the baby is due as the sooner I start it the sooner it will run out.

Thanks