It sounds fine to me. 'Intern' used to be a specifically American term, but it is now used increasingly elsewhere in the English-speaking world. Law students in Britain, for example, need to complete internships as part of their training. And yes, it is fine as verb too. The English language is, and always has been, famously accommodating and flexible. An 'intern' is different from a trainee or an apprentice or a summer temp, and so we develop/absorb this new word to fill the gap in meaning. We also accept words functioning as different parts of speech quite happily too : if 'intern' works as a noun, then we're happy for it to work as a verb as well. And note that the stress is on the first syllable - INtern. If the stress is on the second, it has a different meaning, as Ben points out.

Yes, intern is a noun or a verb. As a verb it means to serve as an intern (the noun form of the word). This means that you can use the verb past tense and say "She interned", which means "She served as an intern".

To sound educated, it would be better to say, "She worked as an intern..." "Intern" as a noun is most often used in American English. In Britain the term is usually "summer trainee". "To intern someone" means to put someone in a war camp as a prisoner of war. These are lax and modern times and some people may accept the use of intern to mean "to work an an intern", but it is risky. Some readers of the sentence/letter may think that the writer is uneducated or sloppy.

I don't disagree with Su.Ki., and I understand alancoots. It is perhaps difficult for della, living in China, where a peasant revolution took place in 1949 on the one hand, and where on the other, students imagine what the West is like from watching American TV series, fully to realise how big a range there is in English usage, and how in certain circles, the forces of tradition are great. For ambitious Chinese students who aim at joining top traditional colleges at the ancient universities (where some undergraduates still form clubs to dine regularly in fancy clothes - full evening tails for men) and top City firms, it is perhaps useful to know the ways of the traditional corridors of power, so that their curriculum vitae and application letters do not offend. Linklaters, the City law firm, for example, has studiously avoided the use of the word "intern" on their web site. There are "summer students", "trainees", and "associates", but no interns, even for their US programmes, though to move with the times they have included the phrase "inclusive culture" on the recruitment page and a female lady of colour in the publicity photo. Students in China, too, need to be given the opportunity to understand the existence and practices of certain extremely prestigious establishments where dark suits and clipped accents reign. My advice to della is: play safe and attune your syntax and vocabulary to the cultural orientation of your target readers.