It has been conjectured by Alspach  that given integers n and m1, . . . , mt with 3 ≤ mi ≤ n and ∑t i=1 mi = ( n 2 ) (n odd) or ∑t i=1 mi = ( n 2 )− n2 (n even) then one can pack Kn (n odd) or Kn minus a 1-factor (n even) with cycles of lengths m1, . . . , mt. In this paper we show that if the cycle lengths mi are bounded by some linear function of n and n is sufficiently large then this conjecture is true.