The frequency dependence of the dipole polarizability α(ω) of P4 is obtained from refractive index measurements on gaseous phosphorus by means of asymmetric white-light interferometry. The experimental results obtained in the range between 10 000 and 30 300 cm-1 yield an extrapolated static value of α=91.71(92)e2a20E- 1H. An extensive theoretical study, based on finite-field many-body perturbation theory and coupled cluster methods, leads to a theoretical estimate at zero frequency of α~93.2(0.9) e2a20E- 1H at the experimental bond length R0=2.1958 Å. For the symmetric stretch of the molecule the dipole polarizability of P4 varies around R0 as [α(R)- α(R0)]/e2a20E- 1H=34.02ΔR+3.68ΔR2- 0.29ΔR3 showing an essentially linear dependence on the change of the P-P bond distance ΔR≡(R-R0)/a0. The dipole polarizability per atom is significantly smaller than that of the phosphorous atom 25.22 e2a20E-1H. This differs from the corresponding change of α in the homologous As4 cluster, where the single atom value is very close to the polarizability per atom of the cluster.