Is it ok to give a one week notice?

This morning I recently received my official start date for a new company that I applied to, however it only allows me to give a 1 week notice to my current employer since my start date is set for 9/22.  I would’ve given a sooner notice but I just wanted to make sure that everything with the new employer cleared before putting in my notice.  I didn’t want to take the gamble of putting in my notice last week and something not clearing with my new employer and not getting the new job offer.

I’ve been with the company for 12 years and they’ve treated me well, but I took a leap of faith on a new opportunity that was brought before me.  I don’t hate my current job and am grateful for everything they’ve done for me, thus why I feel guilty and bad for only being able to provide a 1 week notice.  What makes me feel a little better is that we just hired a new analyst which will be able to back fill my position, but she just started this week.

Anyone been in a similar situation?