Giving a notice sitiation...New job?

Here is my situation. I received a call from a potential new employer HR dept. They offered me a conditional offer for employment, and the official offer will come after the background/reference check and drug test comes back complete. She sent me the e-mail with the link to electronically authorize the reference/background check, and scheduled a drug test date (im doing it tuesday morning, due to monday is labor day).

She said that new orientation training will begin on Sept 23rd.

I've never given a notice to current employers until i receive and official offer and sign papers from the new company.

I know sometimes background/reference checks take time to complete, but if it does take a lil more time than expected, will this be a situation where I could give a 1 week notice instead of 2?

I want to give my current job a 2 weeks notice, b/c I have no personal issues with any of my co-workers and my supervisor. Morally, I dont want to leave them high and dry.

If i can, i will give them a 2 weeks notice. But for any reason I cant, will a 1 week notice be justifiable?