When the 9th of Ab falls on Shabbat, we delay the observance of Tisha B’Ab until the next day, the 10th of Ab, such that Tisha B’Ab begins on Mosa’eh Shabbat. We do not observe Tisha B’Ab earlier, before Shabbat, because this day commemorates the calamity of the Temple’s destruction, and we do not wish to bring in this commemoration sooner. We prefer delaying it in the hopes that in the meantime Mashiah will arrive and there will no longer be a need to observe this day of mourning.
In such a case, we do not recite Habdala on Mosa’eh Shabbat, since we obviously cannot drink wine. Instead, we recite only the Beracha over the candle ("Boreh Me’oreh He’esh") on Mosa’eh Shabbat, and delay the rest of Habdala until Sunday night, when Tisha B’Ab ends.
As Habdala is not recited on Mosa’eh Shabbat in this case, it is especially important to remember to include "Ata Honantanu" in the Amida of Arbit on this night. Normally, if one forgets to recite "Ata Honantanu," he does not repeat the Amida, because he will be reciting Habdala over a cup of wine. When Mosa’eh Shabbat is Tisha B’Ab, however, one does not recite Habdala over a cup of wine that night, and thus his only Habdala recitation which allows him to perform Melacha is "Ata Honantanu." One should therefore make a special effort not to forget "Ata Honantanu" in Arbit on this night.
If one did forget to recite "Ata Honantanu" in this case, he must remember to recite "Baruch Ha’mabdil Ben Kodesh Le’hol" after Arbit so that he can perform Melacha (activity forbidden on Shabbat). According to the view of the Rashba (Rav Shlomo Ben Aderet of Barcelona, 1235-1310), if one forgot to recite "Ata Honantanu" in Arbit on Mosa’eh Shabbat, and then performed Melacha without reciting "Baruch Ha’mabdil," he must repeat the Amida. The Rashba maintained that this case is Halachically equivalent to the case of one who forgot to recite "Ata Honantanu" and then ate before reciting Habdala, upon whom the Sages imposed a "penalty," requiring him to repeat the Amida. In the case of one who performed Melacha, too, according to the Rashba, this penalty is imposed, and the individual must repeat the Amida. The Rambam (Rav Moshe Maimonides, 1135-1204) disagreed, and ruled that this penalty applies only in the case of one who ate. In light of this debate, several authorities (the Bi’ur Halacha, and Hacham Bension Abba Shaul) applied the rule of "Safek Berachot Le’hakel" – that we do not recite a Beracha if there is a question whether it is required – and thus a person who forgot "Ata Honantanu" and then performed Melacha without reciting "Baruch Ha’mabdil" does not repeat the Amida. Hacham Ovadia Yosef, however, ruled that the person in this case should repeat the Amida but have in mind that according to the view of the Rambam, the repetition of the Amida should be considered a Tefilat Nedaba (voluntary prayer).
Truth be told, this question is not likely to arise as a practical matter, because it is generally customary when Tisha B’Ab begins on Mosa’eh Shabbat to recite "Baruch Ha’mabdil" at home when Shabbat ends, and then change one’s clothing and put on non-leather shoes for Arbit. Practically speaking, then, most people recite "Baruch Ha’mabdil" before even coming to the synagogue for Arbit, and so even if one forgets "Ata Honantanu" in Arbit, he may perform Melacha. Regardless
In any event, it is certainly advisable to recite "Baruch Ha’mabdil" before Arbit on this night, and to ensure that one’s wife and children also recite "Baruch Ha’mabdil" at the conclusion of Shabbat so they can perform Melacha.
Summary: When Tisha B’Ab begins on Mosa’eh Shabbat, it is advisable to recite "Baruch Ha’mabdil Ben Kodesh Le’hol" at the conclusion of Shabbat, before Arbit, and this is, indeed, the common practice. If one did not recite "Baruch Ha’mabdil" before Arbit, and he forgot to include "Ata Honantanu" in Arbit, he must ensure to recite "Baruch Ha’mabdil" after Arbit before performing Melacha. If one forgot "Ata Honantanu" and then performed Melacha without reciting "Baruch Ha’mabdil," he must repeat the Amida, and stipulate that according to the view that the repetition is not required, he recites the Amida as a voluntary prayer.
