"The Aleph" begins with the narrator, Borges, recalling that on the February morning of Beatriz Viterbo’s death, a billboard advertisement in the Plaza Constitucion was being changed. The observation prompts him to vow not to allow himself to be changed by life, and thus he intends to consecrate himself to the memory of his beloved. Every year on Beatriz’ birthday (April 30), Borges returns to the house of her father and her cousin, Carlos Argentino Daneri. Whereas in the past it had been necessary to devise pretexts for his visits, the new circumstances permit him to carry on his devotion to Beatriz while seeming to perform an act of courtesy and respect. With each successive visit, he arrives later and stays longer. Gradually he gains the confidence of Carlos Argentino.
Beatriz’ cousin is in effect her antithesis: Beatriz possessed an ethereal quality that almost transcended reality, while Carlos Argentino is altogether too human, as is suggested by his entirely ordinary physical presence and his pointless existence. His remarks about modern man (that it is unnecessary for him to travel since the advent of the telephone, telegraph, radio, cinema, and so forth) cause Borges to make a connection between Carlos Argentino and literature: Both are equally inept, pompous, and vast. When Borges asks why Carlos Argentino does not write down his ideas, Carlos Argentino predictably responds that he has, in a poem entitled "The Earth." As Carlos Argentino reads and comments on the verses, the consummate mediocrity of the work becomes evident. The real task involved in the poem, decides Borges, has been, not its composition, but rather the invention of reasons to explain why it is so admirable. Carlos Argentino’s purpose in his encyclopedic enterprise is to versify the face of the earth, which he does in a boring, unskillful, and chaotic way.
Several Sundays after Borges hears about the poem for the first time, Carlos Argentino unexpectedly requests a meeting between the two, which is to take place in the establishment of Zunino and Zungri. After censuring critics and the practice that he refers to as "prologuemania," Carlos Argentino comes to the point of the interview: He would like Borges’ assistance in securing a prologue for his poem from Alvaro Melian Lafinur, a literary figure of renown. Borges agrees to help, but without any actual intention of doing so.
The following October, the protagonist receives the second call that he has ever received from Carlos Argentino, who this time is extremely upset because Zunino and Zungri are about to demolish his house in order to expand their business. Borges thinks that he understands the cousin’s consternation until Carlos Argentino explains the real cause of his distress: He cannot finish the poem without the Aleph (one of the points in space which contains all points), which is located in the basement.
An accidental discovery of his childhood, it now serves as the source of material for the poem.
Borges tells Carlos Argentino that he will be over immediately, and, upon hanging up, it occurs to him that Carlos Argentino is insane, which would account for his seemingly inexplicable behavior. Once in the house, Borges engages in a conversation with a large portrait of Beatriz until he is interrupted by Carlos Argentino, who is obsessed with the idea of losing the Aleph. After a glass of wine and some instructions about how to view the phenomenon, Borges goes down to the basement and settles into the position that Carlos Argentino has designated. Just when he fears that he has been poisoned and buried by a madman, he sees the Aleph, a small, iridescent sphere of nearly intolerable brilliancy, which reveals the inconceivable universe. He gropes for some emblem or image which might enable him to communicate the experience of witnessing the ineffable, infinite, and vertiginous Aleph, but at the same time he fears contaminating his report with the falseness of literature. Also, he feels despair in the knowledge that the successive nature of language makes it an inadequate vehicle for conveying the simultaneity of spectacle that is the Aleph.
Overcome by feelings of veneration and pity, Borges is once again rudely interrupted by the voice of Carlos Argentino. At that moment, he plots his revenge: He refuses to talk about the Aleph with Carlos Argentino and recommends repose in the serenity of the country. Yet despite his pretense, Borges has indeed been transformed by the experience: All faces seem familiar and nothing surprises him; all is return. Fortunately, however, memory yields to forgetfulness after a few nights of insomnia.
A postscript from the first of March, 1943, updates the report. Carlos Argentino’s poem has won the Second National Prize for Literature, something which Borges ascribes to misunderstanding and envy. Finally, he poses some questions about the Aleph: Did Carlos Argentino choose that name for the sphere, or did he see it used elsewhere applied to the same sort of phenomenon? If, as Borges suspects, the latter is the case, does a genuine Aleph exist, and did he see it and forget it?