The attack came just hours after another suicide bomber targeted a bus carrying Afghan troops in the suburbs of the city, killing at least six soldiers.
The Afghan capital has been hit by a series of deadly Taliban attacks in recent weeks, highlighting the fragility of the security situation as NATO combat forces leave after more than a decade of war.
Police chief Abdul Rahman Rahimi told reporters the bomber blew himself up in the audience watching a theatre show at the Istiqlal High School which is attached to the French cultural centre.
“A suicide bomber blew himself up among the audience in Istiqlal High School (French Cultural Center),” Afghan interior ministry spokesman Sediqi Sediqqi told AFP.
Deputy Interior Minister Mohammed Ayub Salangi said at least one person was killed and 15 others were wounded in the attack.
The toll was confirmed by another police official.
The cultural centre is located in the centre of Kabul, not far from the presidential palace and shares its grounds with the Istiqlal school, a French-financed institution that has taught generations of Afghan children.
Originally opened in 1970, the cultural centre was forced to close between 1983 and 2002 as Afghanistan was torn apart by a series of wars.
It reopened in 2003 and was revamped in 2010. The growing spate of deadly attacks in Kabul in recent weeks has heightened concerns that Afghanistan could tip into a spiral of violence as the US-led military presence declines.
Militants have targeted foreign guest houses, embassy vehicles, US troops and Afghan army buses in Kabul over the past month, undermining claims that the insurgency is weakening as Nato’s 13-year war ends.
US Defense Secretary Chuck Hagel said on a visit to Kabul at the weekend that the upsurge showed “that the international community must not waver in its support for a stable, secure, and prosperous Afghanistan”.
The earlier attack, claimed by the Taliban, saw a bomber on foot target a bus carrying Afghan army troops on the outskirts of the heavily-guarded city.
Afghanistan’s chief executive Abdullah Abdullah warned that Western forces were leaving prematurely in an interview with Britain’s Sunday Times newspaper at the weekend.
Thursday’s attack came as Washington announced it no longer holds any prisoners in Afghanistan, a day after a damning US Senate report on the brutal treatment of “war on terror” detainees.
The Senate report highlighted abuse at “black sites” around the world, several in Afghanistan including once known as the “Salt Pit” outside Bagram Air Base.
Afghan President Ashraf Ghani gave a stern condemnation of the CIA torture detailed in the report, saying it violated “all accepted norms of human rights “and were part of a vicious cycle of violence.