During the hijrah of rasulullah (salallahu alayhi wa sallam), he was with Abu Bakr in the cave while the mushrikeen sought him out. Unable to find him, they placed a significant bounty on his head.
One particular bounty hunter managed to find rasulullah and Abu Bakr; but whenever he would approach them, his horse sank down into the sand and wouldn't move. This website states:
After a few moments the search parry decided that no one could have entered the cave recently, or the spider's web would not have been complete and the dove would not have nested there, and so they left without searching inside. Three days later the Prophet (pbuh) and Abu Bakr thought it safe to leave the cave. Abu Bakr's son, 'Amir, had arranged for three camels and a guide to help them continue their journey to Yathrib. 'Amir would ride behind his father.
The leaders of Quraysh, meanwhile, returned to Mecca and offered a reward of one hundred camels to whoever captured the Prophet (pbuh). Among those who went in search of him was a famous warrior. He was, in fact, the only one to catch up with him, but whenever he came close, his horse would suddenly sink up to its knees in the sand. When this had happened three times, he understood that the Prophet (pbuh) was protected by a power stronger than anything he had known, and so he went back to Mecca. On arriving there he warned everyone against continuing the search, relating what had happened to him.
I believe, although I cannot find any information about it currently, that this particular bounty hunter accepted Islam.
To avoid making this question too broad, I would like to know, particularly, what rasulullah told him after he accepted Islam, and his name (fulan ibn fulan) and clan.
Anything else is bonus.