We study the spectrum of the Jacobi matrix \((\delta_{m,n+1}+\delta_{m,n-1}+aq^ n\delta_{m,n})\), \(m,n=0,1,..\). and the corresponding orthogonal polynomials. The spectral measure is computed when \(q\in (-1,1)\) and sufficient conditions are given to guarantee the absolute continuity of the spectral measure. When \(q>1\) or \(<-1\) the measure is purely discrete. The case \(q=-1\) leads to a set of polynomials orthogonal on the union of two disjoint intervals. When \(q=1\), the polynomials are essentially the Chebyshev polynomials \(\{U_ n(x)\}\).