We study finite-size effects for the gap of the quasiparticle excitation spectrum in the weakly interacting regime one-dimensional Hubbard model with on-site attraction. Two types of corrections to the result of the thermodynamic limit are obtained. Aside from a power law (conformal) correction due to gapless excitations which behaves as 1/Na, where Na is the number of lattice sites, we obtain corrections related to the existence of gapped excitations. First of all, there is an exponential correction which in the weakly interacting regime (|U|≪t) behaves as ~exp(-NaΔ∞/4t) in the extreme limit of NaΔ∞/t≫1, where t is the hopping amplitude, U is the on-site energy, and Δ∞ is the gap in the thermodynamic limit. Second, in a finite-size system a spin-flip producing unpaired fermions leads to the appearance of solitons with nonzero momenta, which provides an extra (nonexponential) contribution δ. For moderate but still large values of NaΔ∞/t, these corrections significantly increase and may become comparable with the 1/Na conformal correction. Moreover, in the case of weak interactions where Δ∞≪t, the exponential correction exceeds higher-order power law corrections in a wide range of parameters, namely for Na≲(8t/Δ∞)ln(4t/|U|), and so does δ even in a wider range of Na. For a sufficiently small number of particles, which can be of the order of thousands in the weakly interacting regime, the gap is fully dominated by finite-size effects.