This paper arose out of an attempt to generalize the q q -1 -basis for the centre of an Iwahori-Hecke algebra q found by Jones to q q -1 and to other types. Considering the Iwahori-Hecke algebra over a subring ξ of q 1/2 q -1/2 , where ξ = q 1/2q -1/2 , we use a new and natural definition of positivity on to describe the "minimal" ξ -basis for Z in terms of a partial order on the positive part of . The main result is to show that this minimal basis is the set of "primitive" minimal elements of the positive part of the centre, for any Weyl group. In addition, the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize (on setting ξ = 0, the equivalent of setting q = 1 in q 1/2 q -1/2 ) to the sum of elements in a conjugacy class, and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class. A constructive algorithm is provided for obtaining the minimal basis. We use the results of Jones to achieve the result in type A n without the need for character theory, and give the result for all Weyl groups using the character theoretical results of Geck and Rouquier. Finally we discuss the non-crystallographic cases and give some explicit examples.