Hi and welcome to English Forums.

We don't write letters for people, but we will review what you write yourself.

Most companies would rather have someone who can commit to being there for a longer period than one month, though, and most will pay. Are those two requirements firm?

They will not care what your requirements are. They want to know that you can do the work.  

Anonymous

“lets see what i wrote

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Dear Sir/Madam

I am a fifth-year student in Environmental Engineering at Pannon University of Veszprém in Hungary. Part of my studies require a one-month internship working in the environmental field.

I am interested in working as an unpaid intern with [name of organization] this summer, but I don't know whether you offer internships. I would greatly appreciate information about the possibilty of an internship with your institute, and if you do have an internship program, more information on the program, including what I should do to apply.

I have enclosed a copy of my Curriculum Vitae and a detailed account of my educational background, but if you need need more information, please do not hesitate to contact me.

I look forward to hearing from you.

Yours faithfully,

”

I always mix up when you use "sincerely" and "yours faithfully" because in the US we use "Sincerely" for everything.

Good luck