Let \(S=R[X,X^{-1};\rho]\) be a skew Laurent polynomial ring over a ring \(R\) with automorphism \(\rho\). The authors characterize those prime ideals \(P\) of \(S\) with \(P\cap R=0\) in terms of irreducible polynomials over the centre of \(Q[X,X^{-1};\rho]\), where \(Q\) is the right Martindale quotient of \(R\) with respect to the filter of non-zero \(\rho\)-invariant ideals of \(R\). The results obtained, which are also applied to the ring \(R[X;\rho]\), are analogous to those given by the second author [J. Algebra 134, 45-59 (1990; Zbl 0702.16002)] on ordinary polynomial rings and by the second author and \textit{J. Matczuk} [Commun. Algebra 18, 689- 710 (1990; Zbl 0711.16004)] on skew polynomial rings of derivation type.