Hadith: 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

عن 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.
[صحيح] - [رواه البيهقي والدراقطني]

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
Sahih/Authentic. - [Al-Bayhaqi]

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."