The 21-year-old father, whose name was withheld in accordance with the law on family violence, broadcast the tragic incident at about 5.45pm in the live clip that was about four minutes long.
The Facebook Live clip drew a lot of viewers and went viral, catching the attention of the mother who rushed to seek help from police, but it was too late.
In the clip, the man was seen sitting with the baby girl on the rooftop of the multi-floor deserted building. He was seen tying a knot around the neck of the girl and dropping her down from the rooftop. She was heard crying briefly.
The man then pulled the baby up and untied the knot. She briefly cried for the last time.
The 21-year-old mother went to Thalang Police Station at about 6.30pm after learning about the clip.
Police searched for a deserted building as seen in the clip and eventually found it located near a five-star hotel at Naithon Beach in tambon Choeng Thalay of Thalang district.
Police found that the man had hanged himself from a beam in the building and the body of the girl was on the rooftop. He did not broadcast his suicide.
The mother said she had a quarrel with the man because he suspected she had a relationship with another man. The woman said she left their house after he checked her phone and threatened to kill her at about 3am on Monday.
Facebook has admitted that broadcasts of murders and suicides have been an issue that it has taken steps to prevent.
"We have a lot of work, and we will keep doing all we can to prevent tragedies like this from happening," Facebook co-founder Mark Zuckerberg said on stage at F8, Facebook's annual developers conference last week, in reference to a video of a murder in Cleveland that was uploaded to Facebook. "Our hearts go out to the family and friends of Robert Godwin Sr," Zuckerberg added in reference to the victim.
On April 16, a Cleveland man uploaded the video of him shooting and killing Godwin, a 74-year-old grandfather who was on his way home from an Easter meal with family.
A TV station was strongly criticized on Facebook for replaying the clip of the baby’s murder. A popular Facebook page, “Drama-Addict”, said although the station blurred the faces, the baby was heard crying.