  We study the rational solutions of the Abel equation $x'=A(t)x^3+B(t)x^2$
where $A,B\in C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$
then the equation has at most two rational solutions. For any other case, an
upper bound on the number of rational solutions is obtained. Moreover, we prove
that if there are more than $(deg(A)+1)/2$ rational solutions then the equation
admits a Darboux first integral.
