The road section, which was still under construction, gave way and crushed part of a bus, killing two and injuring a further 22, with further fatalities still thought to be possible after a car was trapped underneath.
Construction works on the viaduct were supposed to have been finalised before the start of the competition, with efforts stepped over the last few days in preparation for next week's World Cup semi-final.
Belo Horizonte has hosted five World Cup games so far and its last-four match on Tuesday could feature hosts Brazil (if they beat Colombia) against either Germany or France.
Police pic of collapsed viaduct in Belo Horizonte. Been here three weeks, have to say I'm not surprised. Very sad. pic.twitter.com/UOfLNrSKl4— Ben Hayward (@bghayward) July 3, 2014
The capital of Minas Gerais is also home to Argentina in this World Cup, with Alejandro Sabella's side based just 10 kilometres from the collapsed overpass, which was supposed to relieve traffic to the Estadio Mineirao.
The tragedy comes just one day after the daughter of a well-known Argentine journalist was killed in a car crash just outside the city.