We study thermal effects at the depinning transition by numerical simulations of driven one-dimensional elastic interfaces in a disordered medium. We find that the velocity of the interface, evaluated at the critical depinning force, can be correctly described with the power law v~Tψ, where ψ is the thermal exponent. Using the sample-dependent value of the critical force, we precisely evaluate the value of ψ directly from the temperature dependence of the velocity, obtaining the value ψ=0.15±0.01. By measuring the structure factor of the interface we show that both the thermally rounded and the T=0 depinning, display the same large-scale geometry, described by an identical divergence of a characteristic length with the velocity ξ∝v−ν/β, where ν and β are, respectively, the T=0 correlation and depinning exponents. We discuss the comparison of our results with previous estimates of the thermal exponent and the direct consequences for recent experiments on magnetic domain wall motion in ferromagnetic thin films.