The cycle of the moon begins with the Molad, or new moon, in which the moon is completely invisible. It then begins to wax until it peaks as a full moon, and the wanes until it is once again no longer visible. The Gemara (Sanhedrin 41b) states that Birkat Halevana is only recited when the moon is in its waxing phase. Accordingly, the Shulhan Aruch (O.H 426:3) rules that it can be recited up to and including the fifteenth of the Jewish month. On the other hand, the Rama (ibid.) points out that the length of the Jewish lunar month is twenty nine days, twelve hours and seven hundred and ninety three parts. If one were to divide this in half to ascertain when the peak is, the result is a little more than 14 days and eighteen hours after the Molad, as measured in Jerusalem. When taking into account that the peak does not include the entirety of the fifteenth day and that the Molad is calculated in Jerusalem, the Rama rules that one may not recite Birkat Halevana on the night of the fifteenth. Although the Shulhan Aruch rules that it may be recited the entire night of the fifteenth, the Ben Ish Hai (Shana Bet, Parashat Vayikra, § 23), the Kaf HaHaim (K.H., O.H, § 426) and Rabbi Ben Zion Abba Shaul (Vol III, ch. 4, § 6), invoking the principle of Safek Berachot Lehakel, rule stringently like the Rama.
Rabbi Ovadia Yosef (Yabia Omer, vol. VI, § 38) relies fully on the opinion of the Shulhan Aruch and says that one may recite Birkat Halevana the whole night of the fifteenth of the Jewish month. Rabbi Haim Benvenisti (Ba’e Haye) says that one may even recite it the next day, and even though this opinion is not the custom, it demonstrates that one may certainly do so on the night of the fifteenth. The Siddur of Rabbi Ya’akov Ibn Tzur also indicates that Birkat Halevana can be recited on the fifteenth.
It should be noted that the night of the fifteenth day refers to the night which follows the fourteenth of the month and leads into the fifteenth.
Summary: One may recite Birkat Halevana the entire night of the fifteenth of the Jewish month.