I need to do an internship abroad. However, I don't know how to reassure the recruiter about the difference of language in my covering letter. There is no difference of culture as I am from France and looking for an internship in the UK.
First, having lived in England and visited France on numerous, there is more difference than you suggest :).
That being said, the cover letter is intended to sell you in a less quantitative (and more qualitative) way than the CV does. It is most beneficial to focus on what you can offer as a candidate (experience with a project related to the companies work in school, a part time job, etc.). As long as you can converse with a Brit at their speed, that will show in your command of the language when writing the cover letter/CV (and subsequently for phone and in-person interviews).
You did not specify the field you are looking at, but in some fields (hospitality, retail, etc.) having bi-lingual employees is actually a benefit to the company. If this is one of those fields, then by all means draw attention to your bi-lingual status. Otherwise it only serves to draw unnecessary attention away from better selling points. Remember - you only have a short amount of time (and limited paper space) to impress the interviewer, so anything that is not necessary should be cut (and it sounds like you believe your English is up-to-snuff enough not to be an issue).
I don't think the cover letter can do much to let the recruiter know much about your abilities in English. They will assume that you got a fluent speaker to proofread it, so writing it on your own as an example of what you can do won't help. (That, plus proficiency in written English is not always a sufficient indicator of the ability to communicate orally.)
I've never been on that side of this particular problem, but I think a recruiter would mainly look for two things at that stage: prior experience, and certifications.
If you have any prior experience of doing an internship, study, holiday work, or living in an English-speaking country, it should definitely be listed in your CV, and you should also mention briefly it in your cover letter (one sentence, no more). A family vacation doesn't count. A school trip might be worth a mention if you can claim that you successfully interacted with native speakers, I'm not sure about that.
If you have a qualification in English such as TOEFL, it would be listed in your CV, but I don't think it should be mentioned in your cover letter unless English proficiency is a key part of the job (maybe in hospitality?). Getting such a qualification may help your prospects; some companies even require that (though often the requirement can be waived if you have prior experience of working or studying in an English-speaking country). Then again, some companies don't care, so depending on where you apply, it may or may not be worth the investment. Mentioning grades is probably not useful as they vary a lot between countries (I don't expect UK recruiters to have any idea whether 15/20 is good or bad).
If you have any contacts at the place where you're applying, ask them what is expected of interns. For example, it might turn out that the company has a strict qualification policy, in which case you should get started on it.
Your cover letter and CV (and insider contacts and recommendations, if applicable) determine whether you get your foot in the door, but English proficiency is unlikely to be a critical worry at this stage (unless the company has a qualification requirement). The time when your English is evaluated is the interview(s).