The Prophet صلى الله عليه وسلم said:
ثلاث دعوات لا ترد: دعوة الوالد، ودعوة الصائم، ودعوة المسافر
"Three du'aa's are never rejected:
1) The du'aa' of the father,
2) The du'aa' of the one fasting,
3) The du'aa' of the traveller."
In another narration, it is reported that he صلى الله عليه وسلم said:
ثلاث لا ترد دعوتهم: الصائم حتى يفطر، والإمام العادل، ودعوة المظلوم يرفعها الله فوق الغمام، يفتح لها أبواب السماء، ويقول الرب: وعزتي لأنصرنك ولو بعد حين
"Three people's du'aa's are never rejected:
1) The one who is fasting, until he breaks his fast,
2) The just ruler,
3) The one who has been wronged.
Allaah سبحانه و تعالى raises it above the clouds, and the doors of the skies are opened for it, and the Lord سبحانه و تعالى says, 'By My Honour and Glory! I will help you, even if it be after some time!'"
The preceding hadeeth shows that the one who is fasting can make du'aa' throughout the day, and his du'aa' will be responded to.
In a third hadeeth, the time just before breaking the fast has been specified, so a person should strive even more at this particular time to be sincere in his du'aa'. The Prophet صلى الله عليه وسلم said:
إن للصائم عند فطره دعوة لا ترد
"Indeed, the one who is fasting has - while he is about to break his fast - a du'aa' that is not rejected."
  Notes:
 Reported by al-Bayhaqi in his Sunan (3/345) and others. Shaykh al-Albaani رحمه الله declared it to be authentic in as-Saheehah, # 1797.
 Reported by at-Tirmidhi (# 3598), who declared it to be hasan. However, Shaykh al-Albaani رحمه الله pointed out that it has a very slight weakness in it in ad-Da'eefah, # 1358.
 Reported by Ibn Maajah (# 1753) and others. Imaam al-Busayri رحمه الله said (2/350), "Its isnaad is saheeh," and al-Haafidh Ibn Hajar رحمه الله declared it to be hasan (see Shaykh Sa'eed ibn 'Ali al-Qahtaani حفظه الله, Shuroot ad-Du'aa' wa Mawaani' al-Ijaabat, Mu'asasat al-Juraysi, Riyadh, 1416 A.H., p. 121). However, Shaykh al-Albaani رحمه الله did not agree with them, and declared it to be da'eef in his al-Irwaa', # 921. It seems that the hadeeth has a very slight weakness in it, and Allaah سبحانه و تعالى knows best.
 Shaykh Dr. Abu Ammar Yasir Qadhi حفظه الله, Du'a - The Weapon of the Believer, pp. 140-141.
Fasting the Month of Ramadaan