  It is shown that for any irrational rotation number and any admissible gap
labelling number the almost Mathieu operator (also known as Harper's operator)
has a gap in its spectrum with that labelling number. This answers the strong
version of the so-called "Ten Martini Problem". When specialized to the
particular case where the coupling constant is equal to one, it follows that
the "Hofstadter butterfly" has for any quantum Hall conductance the exact
number of components prescribed by the recursive scheme to build this fractal
structure.
