A person got engaged to a girl and the engagement was made public and she put on the engagement ring, but the wedding contract has not yet been concluded because the girl is too young.
Is it permissible for him to sit with her?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

The fiancée, before concluding the wedding contract, is a non-Mahram (marriageable) to the fiancé, and it is impermissible for him to meet her in private or sit with her alone unless in the presence of one of her Mahrams (non-marriageable relatives), and on the condition that she should be dressed according to the Islamic dress code. On the other hand,  the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, said: "No man should meet a woman in private unless in the presence of a Mahram." [Al-Bukhari and Muslim]

It is worth mentioning that putting the engagement ring on the hand of each other before concluding the wedding contract is an unlawful custom, for it involves imitation of the disbelievers, and lets them touch each other, which is unlawful before marriage.  So, a Muslim has to abide by the etiquettes of Islam, and establish his house, from the first day, on piety rather than on violation of the Sharee‘ah (Islamic law).

Allaah Knows best.