Should I quit my current job before I have signed a contract with my new employer?

Anonymous
I got a job offer on the Tuesday just gone with a starting date of March 16th, the lady who does contracts is away this week but I have verbally agreed to salary and start date with the manager. I don't believe my referees have been contacted yet.
I am still in the probation period of my current job and 2 weeks notice would mean I would have to give it tomorrow or Monday.
Should I resign in good faith of the other offer, or wait until I have signed the contract?
Should I quit my current job before I have signed a contract with my new employer?
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