Emergency services were called to Liberton High School in Edinburgh shortly before 10am today.
Keane, who has been named locally and through social media, was pronounced dead at the scene.
Police and health and safety officials have launched an inquiry.
Andy Gray, head of schools and community services at the City of Edinburgh Council said: "This is obviously a very distressing time for all the pupils, the staff, the community of Liberton High School.
"We're offering welfare support to anyone who needs it and we're co-operating fully with Police Scotland and the Health and Safety Executive at this very difficult time."
Superintendent Matt Richards said: "Inquiries now continue with partners from the Health and Safety Executive, and also alongside the council to make sure we get to the bottom of what has happened.
"It would be wrong to speculate about that at the moment. What I will say obviously is that our thoughts and prayers are with the family, and in particular with the school children today and hopefully they can get back to school as soon as possible."
A statement released by Police Scotland said the incident happened within the school's PE block.
Scotland's First Minister Alex Salmond wrote on Twitter: "Shocked and deeply saddened at news from Liberton High. Visited school in December - thoughts today with family, pupils and staff."
The incident comes just weeks after Edinburgh City Council was fined for an incident at the same school after a girl was seriously injured as she fell down a broken-down lift shaft.
One local councillor described today's incident as "tragic and troubling" and said serious questions need to be asked about why it happened.
A number of other parents congregated at the school gates after news of the tragedy emerged and they began collecting their children from the premises from around 12.30pm. They said it has now been closed for the day.
Shailesh Shrestha was at the school to collect his son, who is in S2.
He said: "I was in a meeting when I heard what had happened and have come in to collect my son. We haven't been told any more about it.
"It is concerning that this has happened. You just don't expect this kind of thing to happen inside a school."
A number of pupils were seen leaving the school, covering their faces with scarves and jackets. Several of them looked distressed. A policewoman was seen laying a floral tribute nearby.
Asked whether the council was aware of any concerns about the wall prior to today's incident, Mr Gray said: "We are not aware of that at this stage. We are working with the police and the Health and Safety Executive to really find out thoroughly what has happened today."
Mr Richards said: "At the moment the Health and Safety Executive and the police will meet tomorrow morning to decide the lead. We're working with the council as well to ensure we get to know exactly what happened."
Edinburgh Southern SNP MSP Jim Eadie, whose constituency covers the school, said: "I am deeply shocked and saddened by this tragic incident, and the thoughts of the entire community are with the family and friends of this young girl."
Scotland's Minister for Learning, Alasdair Allan said: "I'm deeply saddened by this terrible news and my thoughts are with the pupil's family and friends.
"We remain in close contact with the council and stand ready to provide any help that may be needed."
Labour MP for Edinburgh South, Ian Murray, and Scottish Conservative leader Ruth Davidson both described the incident on Twitter as "dreadful news" and said their thoughts were with the family and all at the school.
Community figures and politicians voiced their shock and sadness in the wake of the death.
Councillor Nick Cook, who represents the Liberton/Gilmerton ward, said: "It's tragic and troubling news out of Liberton High School.
"Our thoughts are first and foremost with the family.
"But serious questions need to be asked as to how this incident managed to come about."
The Church of Scotland's associate minister at Liberton Kirk, Reverend Dave Rankin, is one of the chaplains at the school.
He said: "Members of the chaplaincy team at Liberton High School went to the school as soon as we became aware of this terrible accident.
"Our thoughts and prayers are with the family of the girl who has died. Ministers and youth workers from a number of local churches were on hand to support both pupils and members of staff as they heard what had happened. The chaplaincy team will offer ongoing support and comfort to the school and the local community as the full impact of this tragedy is felt in the coming days and weeks."
The school's website describes it as a comprehensive school serving the south of Edinburgh. It has 65 staff and 650 pupils.
The latest incident is the second tragedy to hit the school in recent months.
Jamie Skinner, 13, a pupil at Liberton High School, died after collapsing on a football pitch in the city during a match in December. The teenager was playing for Tynecastle FC when he collapsed on the Saughton Astro pitch in Edinburgh at the end of last year.
Edinburgh City Council was fined £8,000 in February after a girl at the same school was seriously injured when she fell more than 16ft (5m) as teachers tried to free her from a broken-down lift.
The schoolgirl, then aged 15, suffered three fractured vertebrae, bruising over her lower back and a sprained wrist as a result.
Edinburgh City Council said the school will remain closed for the rest of this week, except for any pupils seeking emotional support or for S4-6 pupils studying for upcoming exams.
A two-week Easter holiday for Scottish schools starts on Friday.