The answer to your question depends on what you mean by "convert" to Judaism. In general, when we talk about conversion to Judaism we are referring to the process of becoming a member of the Children of Israel. Such a person is obligated to follow all of the Mosaic laws as he is considered the same as any other Jew.
However, I have personally long distinguished between the Jewish religion and the Jewish nation. What we consider "conversion" nowadays involves joining the Jewish nation, which requires the observance of all 613 commandments. But one could convert to accepting the Jewish system of beliefs without actually joining the Jewish nation.
For example, suppose a practicing Christian realized that he really doesn't believe in the Christian understanding of God, but actually the Jewish understanding seems right. He does some studying and realizes the truth of the Torah and the Jewish religion. There is no requirement that this person join the Jewish nation and follow all 613 commandments. Instead, he could choose to remain a gentile and follow the seven Noachide laws, and he would be fulfilling everything that is expected of him. According to Rambam, such a person would merit entering the world to come. (See Mishneh Torah, Sefer Melachim uMilchamot 8:11.