We develop a numerical method based on Sinc functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc L of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredholm integral equation by means of Nyström’s method based on a Sinc quadrature rule. We approximate the matrix and right-hand side of the resulting linear system by an efficient method of computing the Cauchy principal value integrals. The error of an N-point approximation converges to zero at the rate
O
(
e
−
c
N
1
/
2
)
O({e^{ - c{N^{1/2}}}})
, as
N
→
∞
N \to \infty
, provided that the coefficients of the CSIE are analytic in a region D containing the arc L and satisfy a Lipschitz condition in D.