It coincides with the first International Dylan Thomas Day in Swansea.
The old exercise book, lying in a drawer for decades, was bought by Swansea University for Â£104,500 last December.
An exhibition on Thursday will allow a maximum of 200 people to see the book.
Prof John Goodby, of Swansea University, described it as the "holy grail" for Thomas scholars and the most exciting discovery related to the poet since his death in 1953.
It is thought Thomas left it at the house of his mother-in-law in the 1930s but it was forgotten about.
It is one of five notebooks used by Thomas - the other four are in the State University of New York at Buffalo.
The notebook will be on display under protective glass but plasma screens will show the whole of its contents.
The poet's granddaughter Hannah Ellis said the crossings out and re-writings in the book showed the teenage Thomas's meticulous approach and experimentation with language.
"It was so important this notebook would be seen by scholars, fans and the general public," she told BBC Wales.
"The other notebooks had just been left in boxes and it was only during the centenary year they came to Wales and were seen by the public."
The exhibition takes place at the university's council chamber and the date coincides with the first performance of Under Milk Wood in New York in 1953.
A "lost" Dylan Thomas notebook has gone on public display for the first time in the late poet's home city.