This question was examined carefully by María del Carmen Pescador del
Hoyo in 'Cuándo y dónde nació Fernando III el Santo', *Revista de
archivos, bibliotecas y museos* vol. 73 (1966) pp. 499–553.
Ferdinand was the middle of five children born to his parents - the
eldest was Leonor, born in the second half of 1198, followed by Costanza
in 1199 or (more probably) 1200 and then Fernando in June or perhaps
early July 1201: Alfonso IX and his second wife were together in Toro on
31 May, then he occurs alone in Benavente on St John's day (24 June)
from where he went to rejoin the queen. They occur together again in
Salamanca on 26 July, and by 5 August their son was already baptised as
he is named with them on that date ("cum filio meo Ferdinando").
The Jesuit historian Gonzalo Martínez Díez in his 1993 biography of
Ferdinand placed his birth on 24 June, which is repeated as probable
here
http://dbe.rah.es/biografias/10090/fernando-iii. However, this
degree of precision is not warranted.
Peter Stewart