The paper is concerned with a weighted \((0; 0,2)\)-interpolation problem. It is shown that for each even natural number \(n\) and arbitrary numbers \((\alpha_{j, n})^n_{j= 1}\), \((\beta_{ j,n })^{n- 1}_{j=1}\), \((\gamma_{ j,n })^{n-1}_{j =1}\), there exists a uniquely determined polynomial \(P_n\) of degree at most \(3n-2\) such that \[ P_n (x_{j,n})= \alpha_{j,n}, \quad j=1, \dots, n, \qquad P_n (y_{j, n})= \beta_{j, n}, \quad j=1, \dots, n-1, \tag \(*\) \] and \[ (e^{-x^2/ 2} P_n (x) )'_{x= y_{j,n}}= \gamma_{j, n}, \quad j=1, \dots, n-1, \qquad P'_n (0)= \sum^n_{j=1} {{\alpha_{j,n} l_{j,n} (0)^2 H_n' (0)} \over {H_n' (x_{j,n})}}, \tag \(**\) \] where the numbers \((x_{j, n})^n_{j=1}\) and \((y_{j, n})^{n-1}_{j=1}\) are the zeros of the \(n\)-th Hermite polynomial \(H_n\) and its first derivative \(H_n'\), and the polynomials \((l_{j,n})^n_{j=1}\) are the fundamental polynomials of Lagrange interpolation with respect to \((x_{j, n})^n_{j=1}\). In the paper an explicit expression for the polynomial \(P_n\) is given in terms of fundamental polynomials of the first and second kind. Moreover, for a suitable class of continuously differentiable functions \(f: \mathbb{R}\to \mathbb{R}\) the term \(e^{-\nu x^2} |P_n (f, x)- f(x)|\) is estimated on the whole real line, where \(\nu> 3/2\) and the polynomials \(P_n (f, \cdot)\) are determined by \((*)\) and \((**)\), taking \((\alpha_{j, n})^n_{j=1}= (f (x_{j,n} ))^n_{j=1}\), \((\beta_{ j,n })^{n- 1}_{j =1}= (f(y_{ j,n }))^{n- 1}_{j=1}\) and choosing the numbers \((\gamma_{ j,n })^{n- 1}_{j=1}\) in a suitable way.