Open Question: Is it wrong to only give a one day notice?

I had a job interview today and feel like I have a good chance of getting the job. The job would mean a LOT more money for me, benefits, and is somewhat related to the career path I'm pursuing in college (I'm also a college student). The job I have now pays me below my state's min wage (legal because it's a work at home job that is based in another state), cut my hours so I only have 10 a week, and is completely unrelated to what I want to do. I don't even list this job on my resume' and do not want a recommendation from them. My supervisor is however, the nicest person on the planet. If I get the job they want me to start right away and because I don't list my current job on my resume they don't know that I have it. Do you think it would be horribly wrong in this case (if I get the job) to leave with little notice (possibly only 1 day/same day)? I'm not a very important member of the team (I work customer service and company has well over 100 agents).