[Man of the Old Hope]

You're right, Lost One.

In Appendix F of The Lord of the Rings, in the section "Of Hobbits", we read:

"The Hobbits of the Shire and of Bree had at this time, for probably a thousand years, adopted the Common Speech. They used it in their own manner freely and carelessly; though the more learned among them had still at their command a more formal language when occasion required. [By which Tolkien means a formal version of Westron, perhaps like that spoken by the great in Rohan and by the folk of Gondor. He does not mean a different language altogether.]

"There is no record of any language peculiar to Hobbits. In ancient days they seem always to have used the languages of the Men near whom, or among whom, they lived. Thus they quickly adopted the Common Speech after they entered Eriador, and by the time of their settlement at Bree some had already begun to forget their former tongue. This was evidently a Mannish language of the upper Anduin, akin to that of the Rohirrim; though the southern Stoors appear to have adopted a language related to Dunlendish before they came north to the Shire."

In manuscript F2, an early version of "The Appendix on Languages" which was edited into the published Appendix F, and which may be found (edited and with comments by Christopher Tolkien) in The Peoples of Middle-Earth (volume 12 of The History of Middle-Earth), Professor Tolkien gives us a more fleshed-out story:

"§22 Among Hobbits [added: now] there are two opinions. Some hold that originally they had a language peculiar to themselves. Others assert that from the beginning they spoke a Mannish tongue [> Mannish tongues], being, in fact a branch of the race of Men. But in any case it is agreed that after migration to Eriador they soon adopted the Westron under the influence of the Dúnedain of the North-kingdom. The first opinion is now favoured by Hobbits [> is favored by many Hobbits], because of their growing distaste for Men; but there is in fact no trace to be discovered of any special Hobbit-language in antiquity. The second opinion is clearly the right one, and is held by those of most linguistic learning. Investigation not only of surviving Hobbit-lore but of the far more considerable records of Gondor supports it. All such enquiries show that before their crossing of the Mountains the Hobbits spoke the same language as Men in the higher vales of the Anduin, roughly between the Carrock and the Gladden Fields.

"§23 Now that language was nearly the same as the language of the ancestors of the Rohirrim; and it was also allied, as has been said above, both to the languages of Men further north and east (as in Dale and Esgaroth), and to those further south from which Westron itself was derived."