The intrinsic coupling between ferroelectricity and ferromagnetism in multiferroic BiFeO<font size="-1">3</font> is studied using first-principles density-functional theory calculations. The spontaneous polarization in multiferroic BiFeO<font size="-1">3</font> is found to be smaller than that in a pure ferroelectric phase which can be obtained by eliminating ferromagnetism artificially, while the total magnetic moment in multiferroic BiFeO<font size="-1">3</font> is lower than that in a pure ferromagnetic phase in which ferroelectric distortions are artificially frozen out. These results clearly show the competing interaction between ferroelectricity and ferromagnetism in BiFeO<font size="-1">3</font>. We further demonstrate that this competing nature in BiFeO<font size="-1">3</font> plays an important role under external loads: The polarization increases and the total magnetic moment conversely decreases with respect to uniaxial tensile strain along the [111] direction. The polarization and the total magnetic moment change in the opposite way under compressive strain. These results indicate that ferroelectricity and ferromagnetism show opposite strain response with each other. This arises from the mutual charge transfer between the [101]-and [101]-directed Bi-O bonds.