In the paper, it is proved that, for any odd n ≥ 1039, there are words u( x, y) and υ( x, y) over the group alphabet { x, y} such that, if a and b are any two noncommuting elements of the free Burnside group B( m, n), then, for some k, the elements u( a k, b) and υ( a k, b) freely generate a free Burnside subgroup of the group B( m, n). In particular, the facts proved in the paper imply the uniform nonamenability of the group B( m, n) for odd n, n ≥ 1039.