Short answer
Yes, in the sense that it stands inside al-Masjid al-Aqsa, which is the whole walled compound, so prayer in it carries the reward of praying in al-Aqsa. But it is a shrine built over the Rock, not the main prayer hall. The main congregational mosque is the Qibli mosque with the grey dome, at the south of the compound.
Scholars explain that al-Masjid al-Aqsa in the Quran means the whole sanctuary, not one building. The Dome of the Rock, built by the Umayyad caliph Abd al-Malik ibn Marwan around 72 AH (691 to 692 CE), is part of that sanctuary.
The four schools agree on this.
Explanation
Many people think "al-Aqsa" is one building. In fact al-Masjid al-Aqsa is the entire walled area on the hill in Jerusalem, often called al-Haram al-Sharif, with its courtyards, gates, domes and prayer halls. The great reward the Prophet ﷺ promised for praying in al-Aqsa covers the whole of it.
The golden Dome of the Rock (Qubbat al-Sakhra) stands near the centre. The Umayyad caliph Abd al-Malik ibn Marwan built it over a large rock, and its building is dated to around 72 AH (691 to 692 CE). It is used for prayer, and in many periods women have prayed there. But it was built as a shrine over the Rock rather than as the main mosque.
The main prayer hall, where the imam leads the Friday prayer, is the Qibli mosque, with the smaller grey or silver dome at the southern end. It stands roughly where Umar ibn al-Khattab prayed and had a simple mosque built when Jerusalem came under Muslim rule in his caliphate.
So both buildings belong to al-Aqsa, and so do the open courtyards between them. For the full guide, with the history and what the Rock is, read Al-Aqsa and the Dome of the Rock. See also Why is Jerusalem important in Islam?
The evidence
Quran, Surah Al-Israa 17:1
بِسْمِ ٱللَّهِ ٱلرَّحْمَٰنِ ٱلرَّحِيمِ سُبْحَٰنَ ٱلَّذِىٓ أَسْرَىٰ بِعَبْدِهِۦ لَيْلًۭا مِّنَ ٱلْمَسْجِدِ ٱلْحَرَامِ إِلَى ٱلْمَسْجِدِ ٱلْأَقْصَا ٱلَّذِى بَٰرَكْنَا حَوْلَهُۥ لِنُرِيَهُۥ مِنْ ءَايَٰتِنَآ ۚ إِنَّهُۥ هُوَ ٱلسَّمِيعُ ٱلْبَصِيرُ ﴿١﴾
Exalted is He who took His Servant by night from al-Masjid al-Haram to al-Masjid al-Aqsa, whose surroundings We have blessed, to show him of Our signs. Indeed, He is the Hearing, the Seeing.
The Quran names the sanctuary as al-Masjid al-Aqsa.
Sahih al-Bukhari 1189
Sahih... لا تشد الرحال إلا إلى ثلاثة مساجد ...
"Do not set out on a journey except to three mosques."
Al-Aqsa is one of the three mosques worth a special journey.
Where the ruling is recorded
- al-Bidaya wa al-Nihaya (Ibn Kathir), Events of the year 72 AH, the building of the Dome of the Rock. Records Abd al-Malik's building over the Rock.
- Fath al-Bari (Ibn Hajar), Commentary on Kitab Fadl al-Salat fi Masjid Makka wa al-Madina, chapter on the virtue of prayer in the mosques of Makkah and Madinah. Explains the hadith of the three mosques.
- al-Uns al-Jalil bi Tarikh al-Quds wa al-Khalil (Mujir al-Din), Description of al-Masjid al-Aqsa and its buildings. Defines the mosque as the whole enclosure.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.