Francis, 88, was admitted to Rome's Gemelli hospital on February 14 with respiratory problems that degenerated into double pneumonia - a serious infection in both lungs that can inflame and scar them, making it difficult to breathe.
The Vatican said on Saturday evening that the pontiff's condition had stabilised, following an "isolated" breathing crisis a day earlier.
"I would like to thank you for the prayers," Francis said in a note released by the Vatican in place of his usual Sunday prayer with pilgrims, which the Pope was not able to lead for the third week running.
"I feel all your affection and closeness and ... I feel as if I am 'carried' and supported by all God's people," the message said.
In the message from the Pope, he also thanked his doctors for their care and prayed again for peace in Ukraine and elsewhere.
"From here, war appears even more absurd," Francis said in the message, which he drafted in recent days from the Gemelli hospital, the Vatican said.
Francis said he was living his hospitalisation as an experience of profound solidarity with people who are sick and suffering everywhere.
"I feel in my heart the 'blessing' that is hidden within frailty, because it is precisely in these moments that we learn even more to trust in the Lord," Francis said in the text.
"At the same time, I thank God for giving me the opportunity to share in body and spirit the condition of so many sick and suffering people."
Francis also met at the hospital on Sunday with Cardinal Pietro Parolin, the Vatican's number two official, and Parolin's deputy, said Vatican spokesman Matteo Bruni, without giving further details about the meeting.
The Pope, who is known to work himself to exhaustion, has continued leading the Vatican during his hospital stay and last met Parolin and the deputy at the Gemelli on February 24.
Earlier on Sunday, the Vatican said in a one-line update that the Pope had rested well overnight.
A full medical update on his condition is expected on Sunday evening.
A Vatican official, who did not wish to be named because he was not authorised to discuss the Pope's health, said on Sunday that Francis was eating normally, moving about his hospital room, and continuing his treatment.
The Pope suffered a constriction of his respiratory airways on Friday, akin to an asthma attack.
However, in a more upbeat tone on Saturday, the Vatican said the Pope's blood circulation remained stable and he did not have an increased white blood cell count, indicating his infection may be abating.
"The Holy Father's clinical condition remained stable," the Vatican said on Saturday, adding that the prognosis was still guarded, meaning he was not yet out of danger.
The Vatican added on Saturday that for a second day running the Pope required non-invasive, mechanical ventilation, alternating between this and "long periods of high-flow oxygen therapy".
Francis has experienced several bouts of ill health over the last two years and is prone to lung infections because he had pleurisy as a young adult and had part of one lung removed.
The Pope has not been seen in public since entering hospital, his longest absence from view since his papacy started in March 2013, and his doctors have not said how long his treatment might last.
