It is the Dean of the College of Cardinals, currently Cardinal Angelo Sodano. This role for the Dean is many centuries old. The present rules for the election of the Pope are mostly found in the Apostolic Constitution Universi Dominici gregis, issued by John Paul II in 1996. It contains the following:
(19) The Dean of the College of Cardinals, for his part, as soon as he has been informed of the Pope's death by the Cardinal Camerlengo or the Prefect of the Papal Household, shall inform all the Cardinals and convoke them for the Congregations of the College. He shall also communicate news of the Pope's death to the Diplomatic Corps accredited to the Holy See and to the Heads of the respective Nations.
Although this refers to the death of the Pope, the same provision applies in the case of resignation:
(77) I decree that the dispositions concerning everything that precedes the election of the Roman Pontiff and the carrying out of the election itself must be observed in full, even if the vacancy of the Apostolic See should occur as a result of the resignation of the Supreme Pontiff, in accordance with the provisions of Canon 333 § 2 of the Code of Canon Law and Canon 44 § 2 of the Code of Canons of the Eastern Churches.
The citations are to the canons describing how a Pope resigns. UDG §38 confirms the role of the Dean in convoking the cardinals.
Angelo Sodano, however, is 85 years old, and will not be taking part in the conclave - only cardinals under 80 at the time the See becomes vacant are electors. He is also restricted from presiding over the General Congregation, which is the daily meeting of all the Cardinals held before the conclave begins. He can still attend, but doesn't have to.
UDG provides that if the Dean is unable to preside over the conclave himself, then his place is taken by the Sub-Dean (aka Vice-Dean). But this is Cardinal Roger Etchegaray, who is 90 years old! So instead, the role falls (UDG §9) to the senior cardinal who is actually able to take part. This will be Cardinal Giovanni Battista Re, because he is a cardinal bishop (first tier of precedence), and has been one for longer than the only other cardinal bishop who is under 80, Cardinal Tarcisio Bertone. (Cardinal Francis Arinze is 80, and Cardinal Jose Martins is 81. The patriarchs of the Eastern Catholic Churches rank just below the other cardinal bishops.)
Nonetheless, the formal announcement of §19 above could still be made by Cardinal Sodano himself, since all other mentions of "Dean" in the document include a proviso about the substitution if the Dean is impeded, and this section just says "Dean". However, §38 allows another cardinal to do it in his name. This situation arose for the two conclaves of 1978, when both the dean (Cardinal Carlo Confalonieri, 85) and the vice-dean (Cardinal Paolo Morella, 83) were above the age of 80. The next-ranking elector, Cardinal Jean-Marie Villot, was the one who presided over the conclave, and he also sent the messages of summons. Interestingly, he was both Secretary of State and Camerlengo at the time, which probably added some weight to the idea that he should be responsible for the summons as well. (This took place under Romano Pontifici Eligendo, 1975, the predecessor to UDG, but the relevant law is basically the same; see §§19 and 39.)
Update: A Catholic News Service report of 26 February 2013 quotes Father Federico Lombardi, Vatican spokesman, saying that the Dean, Angelo Sodano, will send out letters of summons on 1 March, including email or fax versions as well as the hard copies.