The elastic stress transfer from the matrix to the embedded fibre in fibre-reinforced composites has been analysed previously when the loading direction is parallel to the fibre axis and the fibre is bonded to the matrix. Stress transfer occurs both at the interface along the fibre length and at the ends of the fibre. However, the boundary condition at the bonded ends is ambiguous, and various assumptions have been made to obtain solutions for this stress transfer problem. To satisfy more rigorously the boundary condition for the bonded ends, a new technique of assuming imaginary fibres in the composite is proposed in the present study. Compared to the previous analytical solution, the present analytical solution bears more physical meaning and is in better agreement with numerical and experimental results.