Hungary-Vatican concordat on diplomatic relations (1990): text
With this tiny concordat, Hungary became the second Warsaw Pact country (after Poland did so on 17 July 1989) to establish diplomatic relations with the Vatican. It states that Church-related issues are to be settled in accordance with both Canon (Church) Law and Hungary's new law on religious freedom. It replaces a pact with the Communist regime which is still being held secret. 
between the Holy See and the Republic of Hungary
(Signed 9th February 1990)
With the intention of officially re-establishing and developing mutual relations of friendship, the Holy See and the Republic of Hungary — through their respective representatives duly authorised for this purpose — have agreed as follows:
1. As of today, diplomatic relations are re-established between the Holy See and Hungary, at the level of a Nunciature, on the part of the Holy See, and of an Embassy, on the part of the Hungarian Republic.
2. The Holy See will be represented by an Apostolic Nuncio in Budapest, while the Republic of Hungary will accredit an Ambassador Extraordinary and Plenipotentiary  at the Holy See.
3. Following the profound political and social evolution which has taken place in Hungary in recent months, the issues concerning the Church are now regulated by the new Code of Canon Law  and by the norms of the new Law on freedom of conscience and religion and on the Churches [January 1990]. 
4. As a result, the two Parties consider the partial agreements reached with the Act signed in Budapest on 15th September 1964 with the attached Protocol and two Appendices  to be obsolete, and therefore declare them revoked.
Particular issues of mutual interest that would require bilateral negotiations will be settled by common accord.
Signed in Budapest, the 9th February 1990 in Italian and Hungarian originals, both being equally valid.
For the Holy See,
The Cardinal Secretary of State
For the Republic of Hungary,
The Prime Minister
Miklós Németh s.k.
Accordo tra la Santa Sede e la Repubblica di Ungheria (Relazioni diplomatiche)
Firmato il 9 febbraio 1990
Translated by Graeme A. Hunter
 The text of the 1964 modus vivendi with the Communists is still being hidden. Even its existence is not acknowledged. Hungary's official MT1 news agency announcement of the 1990 concordat, reported in this AP article, only mentioned "a 1950 accord that limited the number of Roman Catholic seminaries and religious orders". Not a word about the secret 1964 agreement.
Presumably the Vatican doesn't want to admit that it was willing to let any state, let alone a Communist regime, control the appointment of its bishops. That could set a precedent. This concession was granted because Cardinal Agostino Casaroli, the Vatican's Secretary of State at the time, was sure that without this 1964 agreement the Hungarian Catholic Church's "complete collapse would have been unavoidable."
Historian Eva S. Balogh describes what is known of this secret pact.
In 1964 there was an agreement between the Vatican and the Hungarian government by which Pope Paul VI agreed to the Hungarian demand that bishops can be appointed only with the permission of the Hungarian state. Unfortunately, historians of state and church relations are totally in the dark concerning the details of this agreement because the Vatican placed all pertinent documents under a seventy-year ban, and in 1998 the Vatican asked the Hungarian government to do the same--not for seventy but for seventy-five years. The Hungarian government (Gyula Horn was the prime minister), in the hope that the Catholic Church would not show outright antagonism to the socialist party and his government, promised whatever the Catholic Church asked. It was a big mistake: the Church openly supported and continues to support Fidesz and Viktor Orbán.
Eva S. Balogh, “The Hungarian Secret Service and the Catholic Church”, Hungarian Spectrum, 29 December 2008. http://esbalogh.typepad.com/hungarianspectrum/catholic-church/
2. “Ambassador Extraordinary and Plenipotentiary” means an ambassador, the highest ranking diplomat who represents his own country to a foreign one to which he is accredited, who is “extraordinary” because, rather than being appointed for a specific purpose, he has a permanent mission and who is “plenipotentary” because he is fully authorized to represent his government.
3. The 1990 Act on the Freedom of Conscience regulates the activities of, and the benefits enjoyed by, religious communities; it also establishes the criteria for legal designation. The county courts implement the registration of churches. The requirements are highly formal: the church must be founded by 100 private individuals and must have a charter and elected organs of administration and representation. The court determines whether the new group complies with constitutional and legal requirements; if so, the court cannot reject the registration request. While any group is free to practice its faith, formal registration grants rights, imposes obligations on operating educational and social institutions, and provides access to several forms of state funding. All registered churches have the same rights and obligations. (Hungary: International Religious Freedom Report 2008)
While this law may grant the more than 360 registered religions the same conditions, the concordat circumvents this. For example, even this first, mini-concordat manages to obligate Hungary to accept Canon Law.
4. Act signed in Budapest on 15th September 1964
As the new Ostpolitik emerged, on the part of the Holy See, negotiations began in May 1963 between the Hungarian Government and the delegation of the Holy See led by Agostino Casaroli, later cardinal secretary of state. As a result of these negotiations, on 15 September 1964, a document (atto) was signed containing a compromise reached by the parties with respect to a small number of issues (procedures for the nomination of bishops, the oath of allegiance to the Constitution required from priests and bishops, and the operation of the Papal Hungarian Institute in Rome), and a much longer protocol, stating the respective positions of the parties on issues where not agreement was reached. The document was not published. It is to be noted that the document cannot qualify as a concordat or a modus vivendi, and not even as an accord, but only as a partial agreement, eventually as an intesa prectica. Representatives of the regime and the Holy See met from that date twice a year, once in Budapest, once in the Vatican, but diplomatic relations could not be re-established.
Balázs Schanda, “Religion and Law in Dialogue: Covenantal and Non-Covenantal Cooperation between State and Religions in Hungary” in Richard Puza and Norman Doe, eds., Religion and Law in Dialogue: Covenantal and Non-Covenantal Cooperation between State and Religion in Europe, 2006, pp. 79-80.