The following should not be relied on as a definitive statement of current law. Only the origin Hebrew version as interpreted by the courts in Israel has final authority. The only way to learn the current Law concerning aliyah (immigration to the State of Israel as a "Jew") is to contact the appropriate authority at your local Israel Embassy or Consulate.
Although I have been most careful in copying out the following, any errors in the text are entirely mine. The original translation from Hebrew into English was made by Dr. Susan Hattis Rolef in "Appendix II" of her book, "The Political Dictionary of Israel", 1987.
2. (a) Immigration shall be on the basis of an immigration visa.
(b) An immigrant visa shall be issued to any Jew who has expressed
a desire to settle in Israel, unless the Ministry of the Interior
is satisfied that the applicant:
(i) acts against the Jewish nation; or
(ii) is liable to threaten the public health or security of the state;
or
(iii) has a criminal past which is liable to endanger the public's
peace.
3. (a) A Jew who has come to Israel and after his arrival expresses
a desire to settle in the country may, while in Israel, obtain an immigrant
certificate.
(b) The exceptions specified in Article 2(b) shall apply to the issue
of an immigrant certificate as well, though a person shall not be regarded
as a threat to public health as a result of an illness which he has contracted
after his arrival in Israel.
4. Every Jew who has immigrated to the country before this law goes into effect and, every Jew who was born in the country, either before or after the law is effective, enjoys the same status as whoever immigrated on the basis of this law.
4A. (a) The rights of a Jew under this law and the rights of an immigrant
under the Citizenship Law - 1952, as well as the rights of an immigrant
under any other legislation, are also imparted on the child or grandchild
of a Jew, and on the spouse of a Jew and on the spouse of the child or
grandchild of a Jew; excluding a person who was a Jew and willingly changed
his religion.
(b) It is immaterial whether the Jew by virtue of whom the right
is claimed under clause (a) is or is not alive and whether he did or did
not immigrate to the country.
(c) The reservations and conditions laid down with regard to a Jew
or immigrant in this law, or under it, or in legislation as mentioned in
clause (a) will apply to those who claim a right under clause (a) as well.
4B. For the purpose of this law a "Jew" - anyone born to a Jewish mother or who has converted, and is not a member of another religion.
The Ministry of the Interior is in charge of the enforcement
of this law and may enact regulations in connection with its implementation
and for the issue of immigrant visas and immigrant certificates to minors
up to the age of 18.
Regulations regarding clauses 4A and 4B require the approval of
the Constitution, Law and Justice Committee of the Knesset.