  In this paper we prove that a smooth embedded paper Moebius band must have
aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth
embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must
converge, up to isometry, to the famous triangular Moebius band. These results
answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and
prove the more specific conjecture of B Halpern and C. Weaver from 1977.
