I think that I would be leary of a company that wont let you give two weeks notice to your current employer.
I also wouldn't give up my current job, until you the people at the new job have agreed to a salary....that's retarded.."quit your old job with no notice so you can't go back...and once we have you in a corner...we'll discuss how much we want to pay you."
The professional thing to do, is to say that out of respect for your current employer, you have to give AT LEAST one week's notice and that perhaps that will give the new employer time to come to an agreement and salary offer BEFORE you agree to work for them. You can also say, that "IF I worked for your company, I'm sure you'd want me to show you the same courtesy, and give you some notice if I intended to leave."
Anyone, that wants you to walk out on your current job with no notice, AND has not committed to you on paper what your salary offer is...isn't someone I'd be in a real big hurry to work for.
Good Luck.
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I can't believe the sheer number of people here that are telling you to walk out with no notice...Apparently common courtesy and respect are dead in today's society...it's really sad.
Show some class.
Pay your current employer the respect of giving them notice.
He may not be able to ensure your future job...but he's given you the job you have had however long you've been there...and he's had the decency to be honest with you about your future with the company.
#1 Establish a solid offer with the new company...no bullllshit "We'll discuss it later"...you need an offer paper BEFORE you agree to take the job.
#2 Give your current boss the common courtesy and decency of at least one week's notice.
#3 Start your new job after the swelling goes down from your wisdom teeth.