"It is a new department in the university."
Translation:Це новий факультет в університеті.
Depends on the context, which country you're talking about, and what you actually mean by "кафедра":
1) If you mean "teachers/lecturers" as in "Вчора кафедра обговорювала навчальний план", then you could use "The faculty was/were discussing the curriculum yesterday".
2) If you mean a sub-department, then it could be "chair". In Germany, where I've been studying, professors had their own "chairs", and such a chair consisted of a professor, 3-4 assistant professors, and around 10 research assistants. Like, for example, "Chair of Strategic and International Management". These people were teaching a set of related courses, pretty much like "кафедри" in Ukraine. I think this is the most direct translation.
3) Finally, if you're talking about a Business School, for example, (which may at the same time be a separate university), then the largest units within such a school will likely be called departments, as you suggested. Like "The Department of Finance at Copenhagen Business School".
There might be other conventions in the US. Here I tried to describe my experience for Europe. So, I don't think there is one correct translation of Ukrainian "кафедра" which would be understood by all English-speakers in the same way. The chioce would depend on whom you're talking to, and which message you want to carry. Hope this helps :)