When you recite Sūrat al-Fātihah, recite "In the name of Allah, the Most Beneficent, the Most Merciful". It is the mother of the Qur'an, the mother of the Book, and the seven oft-repeated verses, and "In the name of Allah, the Most Beneficent, the Most Merciful" is one of these verses
Authentic hadithعن أبي هريرة رضي الله عنه قال: قال رسول الله صلى الله عليه وسلم : إذا قرأتم: الحمد لله فاقرءوا: ﴿بسم الله الرحمن الرحيم﴾ إنها أم القرآن، وأم الكتاب، والسبع المثاني، و﴿بسم الله الرحمن الرحيم﴾ إحداها.
Abu Hurayrah (may Allah be pleased with him) reported that the Messenger of Allah (may Allah's peace and blessings be upon him) said: "When you recite Sūrat al-Fātihah, recite 'In the name of Allah, the Most Beneficent, the Most Merciful'. It is the mother of the Qur'an, the mother of the Book, and the seven oft-repeated verses, and 'In the name of Allah, the Most Beneficent, the Most Merciful' is one of these verses
Explanation
This noble Hadīth indicates that it is permissible to recite the Basmalah at the beginning of Sūrat al-Fātihah in prayer, because it is part of the Sūrah. However, he means that it should be recited in a low voice and not loudly, since this is what has been transmitted in more authentic Hadīths. At-Tahāwi said: "Abandoning reciting the Basmalah aloud in prayer has been collectively and contiguously reported from the Prophet (may Allah's peace and blessings be upon him) and his successors."
Source · Al-Bayhaqi - Narrated by Ad-Daraqutny
From the Hadith Encyclopedia (hadeethenc.com).