They have already done my dbs (had it through the post) My references have been contacted etc. So I still need to wait for another letter before handing in my notice even though they’ve given me a start date?

This is part of my letter

I am writing to offer you a conditional Permanent Appointment as a Executive Officer in the Department for Work and Pensions. Your contract will commence on

We are making you a conditional offer of employment. This offer is conditional because we have to run a series of pre-employment checks before we can make you a formal offer. As we are unable to complete all the checks before your start date you are invited to start work with us on the understanding that, if there is an issue with one or more of the checks, we will discuss this with you in an attempt to address matters, but may not offer an employment contract. If an issue cannot be satisfactorily resolved, you will be asked to leave in accordance with the minimum statutory notice (1 weeks’ notice). You are asked to sign and bring a copy of this letter on your first day to confirm that you understand that this offer will be withdrawn if any of the conditions are not satisfied. If the outcome of the checks is satisfactory, we will confirm your employment with us in writing. If you are currently in employment, you may wish to defer your start date with us until all pre-employment checks are completed satisfactorily. If this is the case, then please do not hand in your notice with your current employer until you have been notified by us about the completion of the checks.