Although the main text of the Hadeeth is authentic and recorded in the Two Sahihs, the addition, "Except in Makkah," is weak. The correct version of the Hadeeth, which is related by a number of Companions reads as follows:
لا صلاة بعد الصبح حتى ترتفع الشمس ولا صلاة بعد العصر حتى تغيب الشمس
"There is no prayer after Subh (Fajr), until the sun rises; and there is no prayer after 'Asr until the sun sets." (Al-Bukhari no. 586)
However, from this general ruling, an exception is made for prayers that are performed for a reason — such as the Eclipse prayer, the prayer after Tawaf, and the prayer one makes upon entering the Masjid - and this is the correct ruling from the two views the scholars hold on the issue. Prayers that are performed for a reason, such as those just mentioned, may - as an exception - be performed during the two forbidden times, because of authentic Hadeeths that point to this ruling. And with Allaah is the facilitation to do what is right.