Let \(U_N=\{u_0,...,u_N\}, (V_N=\{1,v_1,...,v_N\})\) be a Chebyshev (Markov) system on the interval \([a,b],\) respectively. For a given set of ordered non-overlapping intervals \([c_k,d_k]\subseteq [a,b], k=1,...,n\) the authors consider the multiple node interval quadrature formula (with respect to \(V_N\)) \[ \int_a^b\mu(t)f(t)dt\approx \sum_{k=1}^n\sum_{\lambda=0} ^{\nu_k-1}\frac{a_{k,\lambda}}{d_k-c_k}\int_{c_k}^{d_k} \mu(t)f(t)v_\lambda(t)dt, \] where \(\mu\) is a positive weight on \([a,b]\) and \(\sum_{k=1}^n\nu_k= N+1.\) This formula is said to be Gaussian with respect to \(U_N\) if it is exact for each \(u\in U_N\) and \(a_{k,\nu_k-1}=0, k=1,...,n.\) Using the antipodality theorem of Borsuk, the authors prove the existence of Gaussian interval quadrature formula. The unicity of Gaussian formula and the description of Gaussian intervals \([c_k,d_k]\) is also presented in some particular cases.