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In current use, both the English verb disappoint and the French verb désappointer mean: to frustrate the expectation or desire of (a person).

The English verb disappoint, which dates back to Middle English, was borrowed from the French verb désappointer, which dates back to Middle French. The primary (and now obsolete) meaning of those verbs was: to undo the appointment of (a person), i.e., to deprive (a person) of an appointment, of an office, etc.—hence, also, by extension: to frustrate the expectation or desire of (a person).

The English lexicographer Randle Cotgrave (died circa 1634) recorded both meanings in A Dictionarie of the French and English Tongues (London: Printed by Adam Islip, 1611) [at cited words]—m. stands for masculine, and f. for feminine:

Desapointé: m. ée: f. Disappointed, frustrated; also, remooued [sic], or put from an office, estate, or authoritie.
Desapointer. To diappoint [sic], or frustrate; also, to remoue, or put from an office, or estate; to depriue, or bereaue of authoritie.

By the second half of the 18^th century, the French verb désappointer had ceased to be used in the sense: to frustrate the expectation or desire of (a person).

It was the French philosopher, playwright, poet, historian and polemicist François-Marie Arouet (1694-1778), known as Voltaire, who revived the use of désappointer in this sense, by referring to the English language, which had not ceased to use disappoint in this sense.

The English lexicographer Samuel Johnson (1709-1784) defined the verb disappoint as follows in A Dictionary of the English Language: In which the Words are deduced from their Originals, and Illustrated in their Different Significations by Examples from the best Writers. To which are Prefixed, a History of the Language, and an English Grammar (London: Printed by W. Strahan, For J. and P. Knapton; T. and T. Longman; C. Hitch and L. Hawes; A. Millar; and R. and J. Dodsley – 1755) [at cited word]:

To defeat of expectation; to balk; to hinder from something expected.

First, Voltaire (who was an Anglophile and an Anglophone) wrote the following in a letter to l’abbé d’Olivet [note 1], dated Château de Ferney, 20^th August 1761—as published in Nouveaux mélanges philosophiques, historiques, critiques, &c. &c. ([s.l.]: [s.n.], 1765) [Volume 3; page 377 to 379]:

Il y a plusieurs anecdotes curieuses, qu’il est agréable de savoir. Il y a plus d’une remarque à faire sur la langue. Je trouve, par exemple, plusieurs mots qui ont vieilli parmi nous, qui sont même entiérement oubliés, & dont nos voisins les Anglais se servent heureusement. Ils ont un terme pour signifier cette plaisanterie, ce vrai comique, cette gayeté, cette urbanité, ces saillies qui échapent à un homme sans qu’il s’en doute ; & ils rendent cette idée par le mot humeur, humour, qu’ils prononcent yumor ; & ils croyent qu’ils ont seuls cette humeur, que les autres nations n’ont point de terme pour exprimer ce caractère d’esprit. Cependant c’est un ancien mot de notre Langue, employé en ce sens dans plusieurs comédies de Corneille [note 2]. […]
[…]
Que d’expressions nous manquent aujourd’hui, qui étaient énergiques du tems de Corneille, & que de pertes nous avons faites, soit par pure négligence, soit par trop de délicatesse ! On assignait, on apointait un tems, un rendez-vous ; celui qui, dans le moment marqué, arrivait au lieu convenu, & qui n’y trouvait pas son prometteur, était désapointé. Nous n’avons aucun mot pour exprimer aujourd’hui cette situation d’un homme qui tient sa parole, & à qui on en manque.

The following translation of the above-quoted passage, by the English clergyman and author Thomas Francklin (1721-1784), is from Letters of M. de Voltaire, to several of his Friends (Glasgow: Printed for Robert Urie, 1770) [pages 11 to 13]:

There are some curious anecdotes worth knowing, particularly with regard to my remarks on the language. I find, for instance, several words grown obsolete amongst us, and even totally forgotten, which our neighbours the English make use of with success. They have a term to signify true comic pleasantry, that gaiety and urbanity, those natural sallies which escape a man even without his own consciousness of them. This idea they express by the word humour, which they pronounce yûmour, and which they imagine is possessed by themselves alone, and that other nations have no term which sufficiently marks out this species of wit: it is notwithstanding an old word in our language, and used in this sense in several of the comedies of Corneille. […]
[…]
How many words do we want now, which had great energy and strength in the time of Corneille, and how much have we lost, either from mere negligence, or too much delicacy! A time or a rendezvous was assigned or appointed; he who arrived at the place agreed on, and did not meet with the persons who had made the promise, was * disappointed. We have no word at present to express the precise situation of a man who keeps his word whilst another breaks it.
[Translator’s note] * It seems rather extraordinary that when Mr. Voltaire was comparing the English and French words together, he should forget our word disappoint [note 3], and not observe, as he has done with regard to the word humour, that Mr. Corneille, and other writers of that time, most probably took it from us.

Then, Voltaire referred to the English verbs appoint and disappoint in Questions sur l’Encyclopédie, par des amateurs ([s.l.]: [s.n.], 1771) [Volume 2; s.v. apointé, désapointé; page 30]:

Les Anglais ont pris de nous ces mots apointé, désapointé, ainsi que beaucoup d’autres expressions très-énergiques ; ils se sont enrichis de nos dépouilles, & nous n’osons reprendre notre bien.
     Translation:
The English have taken from us those words appointed, disappointed, as well as many other very vigorous expressions; they have grown rich with our spoils, and we dare not claim back our property.

From the first edition (1694) to the fifth edition (1798) of the Dictionnaire de l’Académie française, the verb désappointer was defined solely in reference to its primary sense: to undo the appointment of (a person). It was only in the sixth edition (1835) of the Dictionnaire de l’Académie française that désappointer was defined in reference to the sense: to frustrate the expectation or desire of (a person). The current ninth edition mentions the fact that Voltaire revived the latter sense by referring to the English verb disappoint.

Notes:
1 This refers to the French clergyman, grammarian, translator and member of the Académie française Pierre-Joseph Thoulier d’Olivet (1682-1768), known as l’abbé d’Olivet.
2 This refers to the French poet and playwright Pierre Corneille (1606-1684).
3 In my opinion, Voltaire did not “forget [the English] word disappoint”, but was implicitly referring to it.