For a fixed $T$ and $k \geq 2$, a $k$-dimensional vector stochastic differential equation $dX_t=μ(X_t, θ)dt+ν(X_t)dW_t,$ is studied over a time interval $[0,T]$. Vector of drift parameters $θ$ is unknown. The dependence in $θ$ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter $\overlineθ_n\equiv \overlineθ_{n,T}$ obtained from discrete observations $(X_{iΔ_n}, 0 \leq i \leq n)$ and maximum likelihood estimator $\hatθ\equiv \hatθ_T$ obtained from continuous observations $(X_t, 0\leq t\leq T)$, when $Δ_n=T/n$ tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on $\hatθ$ and on path $(X_t, 0 \leq t\leq T)$. The uniform ellipticity of diffusion matrix $S(x)=ν(x)ν(x)^T$ emerges as the main assumption on the diffusion coefficient function.