Let \(S\) be a rational surface of \(\mathbb P^3\), the complex projective space. Assume that a parametrization of \(S\) is assigned, given by four homogeneous polynomials \(P_i\), \(1\leq i\leq 4\), of degree \(r\) in variables \(x\in\mathbb C^3\). The authors propose a new algorithm to eliminate the variables \(x\), which is based on the classical Sylvester's resultant and precisely on a method, described by Salmon in 1885, to compute the resultant of three plane algebraic curves of the same degree. This algorithm, called ASSIA (Adapted Sylvester Surface Implicitization Algorithm), applies only under the assumption that the polynomials \(P_i\) never vanish simultaneously. If the parametrization is birational, the algorithm actually produces the equation of \(S\), otherwise it gives a suitable power of this equation. Several examples are given. It is interesting to note that, for effective computations, this method is preferable to the approach based on Gröbner bases.