Can I apply to medical school with marks from a one-year degree?

I would think that they’d look at all your marks as a whole but there’s no harm in submitting it with a fabulous letter as to why they should accept you. The worst thing you’ll get in return is “declined”.

Nothing ventured; nothing gained.

Whast is a one-year bachelor’s degree? In what concentration and from what institute?

They will ask for all of your transcripts.

Is there such thing as a one year bachelor’s degree?

Yeah – what is that?

If you mean you went to a post-baccalaureate program (to pick up additional sciences to improve your chances of being accepted to medical school) – yes, the medical school admissions board will pay a bit more attention to your post-baccalaureate program grades (esp. if they were courses such as biochem, chem and other studies directly related to a pre-med education). They will also look at your other undergraduate degree and grades.
If there is a significant difference between the two (for example, your original B.A. you had only a 3.1, and your post-bac certificate you got a 4.0) expect to explain the difference in your performance in interviews with schools. They will no doubt ask whether you failed to apply yourself in your original 4 year program or not.

Bear in mind that to admissions, usually, GPA (for either your original degree or the one year program) are less important than:
1) life experience (if you’ve spent years working as an EMT or nurse for example, this will be a strong positive)
2) Your MCAT scores

Right now entry into (good) medical schools is as tough as it’s ever been—so what will differentiate you as a candidate will be something unique (life experience, past work history,multilingual in 2 or more languages) or something that shows you are a strong student (and grades in a one-year post-baccalaureate program don’t mean strong student..unless that program was an extremely challenging program such as the one at Univ. of Chicago or another highly competitive school.)

Any medical school worth attending will require you to submit all of your transcripts. If you send only your post-baccalaureate records, they will know that they are missing something and will consider your application to be incomplete.