If after this second warning they continue to illegally download copyrighted content, the internet service provider will cut off access to the internet for a year.
The legislation passed with a massive majority in France, the EUobserver.com web site reported. However, it still needs approval by the lower house before it becomes French law.
The European Parliament voted in favour of the so-called "Telecom Packet" last July. Although the law itself is meant to regulate the European telecommunications market, the amendments to the law target online piracy.
The bill was also accused by other French senators of being too severe, pointing out that often internet access comes bundled with television and fixed-line telephone services and that it is impossible to just cut off the internet.
Privacy rights organizations do not agree and argue that the law disregards privacy rights. The bill had been said to lead to a "Soviet internet" and that it would make spying a natural obligation for communications providers.