She formed the intention to make up a missed Ramadan fast as the adhaan for Fajr began; is her fast valid?

Praise
be to Allah.

Firstly: 

Having the intention from the night before is a condition for every
obligatory fast, according to the more correct scholarly opinion, whether
that is making up that missed fast or observing it on time. This is the view
of the majority of scholars. 

Ibn Qudaamah said: If it is an obligatory fast, such as a Ramadan fast being
observed on time or being made up later, or a fast in fulfilment of a vow,
or a fast offered in expiation, it is stipulated that the person should form
the intention from the night before, according to our imam, and Maalik and
ash-Shaafa‘i. Abu Haneefah said: It is acceptable for a Ramadan fast and any
fast for a specific reason to form the intention from that day.

End quote from al-Mughni (3/109) 

The evidence for it being obligatory to form the intention from the night
before is the report which was proven to be from the Prophet (blessings and
peace of Allah be upon him), according to which he said: “Whoever does not
decide to fast before dawn, his fast does not count.” Narrated by at-Tirmidhi
(730) and classed as saheeh by al-Albaani in Saheeh at-Tirmidhi. At-Tirmidhi
said, after quoting this hadith: What this means, according to the scholars,
is that there is no fast for the one who does not decide to fast before dawn
breaks, whether in Ramadan or when making up missed Ramadan fasts or when
fasting in fulfilment of a vow. If he does not form the intention from the
night before, then it does not count. As for voluntary fasts, it is
permissible for him to form the intention after he gets up (in the morning).
This is the view of ash-Shaafa‘i, Ahmad and Is-haaq. End quote. 

Secondly:

The one who wants to observe an obligatory fast must form his intention
before the true dawn breaks, as Allah, may He be exalted, says
(interpretation of the meaning):

“and eat and
drink until the white thread (light) of dawn appears to you distinct from
the black thread (darkness of night)”

[al-Baqarah 2:187]. 

So
what counts is the break of dawn, not the adhaan. If a person is certain
that the true dawn has broken, and he did not form the intention to fast,
then his observance of an obligatory fast, whether he was doing it on time
or was making it up, is not valid. 

But if he is not certain that dawn has broken, then he may delay forming the
intention until the last moment before dawn does break. The same applies if
he knows that the mu’adhdhin gives the adhaan ahead of time, or he is not
sure whether he gives the adhaan on time or ahead of time. 

Please see the answer to question no. 66202

Thirdly: 

Most mu’adhdhins nowadays rely on clocks and timetables, not on actually
sighting the dawn, and this cannot be regarded as certainty that dawn has
broken. So whoever eats or forms his intention at that time, his fast is
valid, especially if he stops eating when the adhaan begins, as in the case
asked about here, because he was not certain that dawn has broken. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: What
is the Islamic ruling on the fast of someone who hears the adhaan of Fajr
but carries on eating and drinking? 

He
replied: What is required of the believer is to refrain from things that
break the fast – eating, drinking and so on – when it becomes clear to him
that dawn has broken, if the fast is obligatory, such as in Ramadan, fasts
observed in fulfilment of vows, and expiatory fasts, because Allah, may He
be glorified and exalted, says (interpretation of the meaning):

“and eat and
drink until the white thread (light) of dawn appears to you distinct from
the black thread (darkness of night)”

[al-Baqarah 2:187]. 

If
he hears the adhaan and knows that the adhaan is usuall given at the time of
dawn, then he must stop eating and drinking. 

If
the mu’adhdhin gives the ahdaan before dawn breaks, then he does not have to
stop eating and drinking; it is permissible for him to eat and drink until
it becomes clear to him that dawn has broken. 

If
he does not know whether the mu’adhdhin gives the adhaan before or after
dawn breaks, then to be on the safe side it is better for him to stop eating
and drinking when he hears the adhaan, and it will not matter if he eats or
drinks something at the time of the adhaan, because he does not know whether
dawn has broken. It is well-known that those who live in cities where there
are electric lights cannot know whether dawn has broken from looking, but to
be on the safe side they should follow the adhaan and timetables which state
the time of dawn in hours and minutes, acting in accordance with the words
of the Prophet (blessings and peace of Allah be upon him): “Leave that which
makes you doubt for that which does not make you doubt” and “Whoever guards
against the doubtful matters will protect his religious commitment from
shortcomings and will protect his honour from slander.” And Allah is the
source of strength. 

End quote from Fataawa Ramadan, compiled by Ashraf ‘Abd al-Maqsood
(p. 201) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: When is it
forbidden for a person to eat? Is it as they say: when the mu’adhdhin says
Laa ilaaha ill-Allah? What is the ruling if he drinks
deliberately after the adhaan? Is he like one who drinks after ‘Asr, or does
his fast count? Some people justify that by saying that dawn is not like a
lamp that begins to shine quickly, and the matter is broad in scope. What is
the ruling? 

He
replied: If the mu’adhdhin gives the adhaan when dawn becomes clear, then
the Prophet (blessings and peace of Allah be upon him) said: “Eat and drink
until Ibn Umm Maktoom gives the adhaan, for he does not give the adhaan
until dawn breaks.” If the mu’adhdhin says: I have seen the dawn and I do
not give the adhaan until I see the dawn, then the individual must stop
eating and drinking when he hears the adhaan, except in the case concerning
which a concession has been made, which is a person has his vessel in his
hand, in which case he may drink what he needs from it. 

But if the adhaan is given according to a timetable, then the timetable is
not actually connected to the actual visible time – as it is based on
calculations. The timetables we have nowadays, such as the Umm al-Qura
timetable and others, are based on calculations, because they do not
actually observe the dawn, or the sun, or the zenith, or the beginning of
‘Asr, or the setting of the sun.

End quote from al-Liqa’ ash-Shahri (1/214) 

For more information, please see the answer to question no.
124608 

Based on the above, your fast is valid, in sha Allah, because we cannot be
certain that the mu’adhdhin was giving the adhaan at the time when dawn
began. 

And Allah knows best.