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The present work studies the combined effects of weak turbulence and intrinsic flamefront hydrodynamic (Darrieus-Landau, DL) instability on the dynamics of premixed flames. Recent modeling [A. Gruber, J.H. Chen, D. Valiev, C.K. Law, J. Fluid Mech. 709 (2012) pp. 516-542] showed that DL instability contributes to the flashback of premixed hydrogen-air flames in the boundary layer of a turbulent channel flow. A concept of a turbulence-induced DL cutoff as a function of the laminar DL cutoff and the turbulence intensity was introduced for high turbulence intensities [S. Chaudhuri, V. Akkerman, C.K. Law, Phys. Rev. E 84 (2011) art. no. 026322]. It was shown qualitatively that increasing turbulence intensity limits the instability development to large scales, thereby moderating the entire effect of the instability. In order to investigate the low turbulence intensity limit in greater detail, an initially planar flame is subjected to oncoming turbulence in direct numerical simulations (DNS) with detailed hydrogen/air chemistry. A parametric study is performed by varying the turbulence intensities and the length scale of the domain. By examining the flame front shape and velocity evolution it is found that for certain ranges of turbulence intensity, the development of the DL instability is significantly affected by turbulence. It is shown that the DL instability of the flame front that is stable in the laminar case may be triggered by weak turbulence.