New job starts in one week. No offer letter. Current job doesn't know--should I give notice?

I have verbal confirmation that I'm starting next Monday at the new company. Their contract was recently tripled in capacity and they're hiring a ton of people last minute to fill their obligations. I was interviewed Thursday with verbal confirmation of onboarding next Monday and an offer letter "very soon". New job doesn't know about my current job.

This Monday, I will be one week away from my new job's start date. Should I give notice to my current job or let that bridge burn?

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