Close encounters of stars to the Sun could affect life on Earth through gravitational perturbations of comets in the Oort cloud or exposure to ionizing radiation. By integrating orbits through the Galactic potential, I identify which of 33 million stars in Gaia DR3 with complete phase space information come close to the Sun. 61 stars formally approach within 1 pc, although there is high confidence in only 42 (two thirds) of these, the rest being spurious measurements or (in) binary systems. Most of the stars will encounter within the past or future 6 Myr; earlier/later encounters are less common due to the magnitude limit of the Gaia radial velocities (RVs). Several close encountering stars are identified for the first time, and the encounter times, distances, and velocities of previously known close encounters are determined more precisely on account of the significantly improved precision of Gaia DR3 over earlier releases. The K7 dwarf Gl 710 remains the closest known encounter, with an estimated (median) encounter distance of 0.0636 pc (90% confidence interval 0.0595-0.0678 pc) to take place in 1.3 Myr. The new second closest encounter took place 2.8 Myr ago: this was the G3 dwarf HD 7977, now 76 pc away, which approached within less than 0.05 pc of the Sun with a probability of one third. The apparent close encounter of the white dwarf UPM J0812-352 is probably spurious due to an incorrect RV in Gaia DR3.