Baryon Acoustic Oscillations (BAOs) provide a "standard ruler" of known physical length, making it one of the most promising probes of the nature of dark energy (DE). The detection of BAOs as an excess of power in the galaxy distribution at a certain scale requires measuring galaxy positions and redshifts. "Transversal" (or "angular") BAOs measure the angular size of this scale projected in the sky and provide information about the angular distance. "Line-of-sight" (or "radial") BAOs require very precise redshifts, but provide a direct measurement of the Hubble parameter at different redshifts, a more sensitive probe of DE. The main goal of this paper is to show that it is possible to obtain photometric redshifts with enough precision (σ _z ) to measure BAOs along the line of sight. There is a fundamental limitation as to how much one can improve the BAO measurement by reducing σ _z . We show that σ _z ~ 0.003(1 + z) is sufficient: a much better precision will produce an oversampling of the BAO peak without a significant improvement on its detection, while a much worse precision will result in the effective loss of the radial information. This precision in redshift can be achieved for bright, red galaxies, featuring a prominent 4000 Å break, by using a filter system comprising about 40 filters, each with a width close to 100 Å, covering the wavelength range from ~4000 to ~8000 Å, supplemented by two broad-band filters similar to the Sloan Digital Sky Survey u and z bands. We describe the practical implementation of this idea, a new galaxy survey project, PAU¹⁶Physics of the Accelerating Universe (PAU): http://www.ice.cat/pau., to be carried out with a telescope/camera combination with an etendue about 20 m² deg², equivalent to a 2 m telescope equipped with a 6 deg² field of view camera, and covering 8000 deg² in the sky in four years. We expect to measure positions and redshifts for over 14 million red, early-type galaxies with L > L and i_AB 22.5 in the redshift interval 0.1 < z < 0.9, with a precision σ _z < 0.003(1 + z). This population has a number density n 10^–3 Mpc^–3 h ³ galaxies within the 9 Gpc³ h ^–3 volume to be sampled by our survey, ensuring that the error in the determination of the BAO scale is not limited by shot noise. By itself, such a survey will deliver precisions of order 5% in the dark-energy equation of state parameter w, if assumed constant, and can determine its time derivative when combined with future cosmic microwave background measurements. In addition, PAU will yield high-quality redshift and low-resolution spectroscopy for hundreds of millions of other galaxies, including a very significant high-redshift population. The data set produced by this survey will have a unique legacy value, allowing a wide range of astrophysical studies.