We discuss the first infrared reflectivity measurement on a BiFeO3 single crystal between 5 K and room temperature. The 9 predicted ab-plane E phonon modes are fully and unambiguously determined. The frequencies of the 4 A1 c-axis phonons are found. These results settle issues between theory and data on ceramics. Our findings show that the softening of the lowest frequency E mode is responsible for the temperature dependence of the dielectric constant, indicating that the ferroelectric transition in BiFeO3 is soft-mode driven.PACS numbers: 78.30.-j, 77.84.-s, 77.22.-d The interplay between magnetic and dielectric properties is an intriguing subject that was already discussed by Pierre Curie in 1894 [1]. Such, magnetoelectric effects were subject of extensive studies during the 1960s [2,3,4]. More recently, the ability of controlling the dielectric properties through a magnetic field and viceversa renewed the interest in these materials [6,7,8]. Coupling between electrical and magnetic properties are particularly interesting in multiferroic materials, which present simultaneously two or more ferroic or antiferroic order parameters. Optical spectroscopy is a very powerful tool to understand the driving mechanism of the ferroelectric transition and, eventually, its coupling to magnetic ordering [10,11]. Nevertheless, only a few Raman [12,13,14] and infrared (IR) [15,16] studies on BiFeO 3 are known to date. Surprisingly none of the IR measurements were carried out on single crystals. There are two important open questions concerning the electrodynamics of BiFeO 3 : (i) the phonon modes symmetry; and (ii) the role (or absence of thereof) of phonon softening in the ferroelectric transition.In its ferroelectric phase, BiFeO 3 belongs to the R3c (C 6 3v ) space group, which derives from the ideal (paraelectric) P m3m (O 1 h ) cubic perovskite group by a small distortion along the (111) cubic directions. Its magnetic structure is a spiral-cycloidal incommensurate phase. Group theory analysis for the R3c ferroelectric group predicts 4A 1 ⊕ 5A 2 ⊕ 9E optical phonon modes [16]. The A 1 modes are infrared active along the c-axis, the E modes are infrared active on the ab-plane and the A 2 modes are silent. Early Raman data [12] found at least 11 modes at room temperature. Kamba et al. [16] proposed to fit the infrared data on ceramics using 13 Lorentz oscillators. Although these results found the proper number of modes, they cannot clarify the symmetry of the phonons. Some extra insight on the phonon assignment was obtained on more recent Raman data [13,14] but some inconsistencies in phonon assignment remain. In addition, Haumont et al. [12] argued that because the frequency of the lowest mode did not vanish smoothly at the Curie temperature, the ferroelectric transition in BiFeO 3 would not be soft-mode driven. This picture was later revised to propose that the incomplete phonon softening could rather be a sign of a first order transition [16,20]. However, both scenarios remain speculative.In this paper, we show the p...