  Let $B^{a,b}:=\{B_t^{a,b},t\geq0\}$ be a weighted fractional Brownian motion
of parameters $a>-1$, $|b|<1$, $|b|<a+1$. We consider a least square-type
method to estimate the drift parameter $\theta>0$ of the weighted fractional
Ornstein-Uhlenbeck process $X:=\{X_t,t\geq0\}$ defined by $X_0=0; \ dX_t=\theta
X_tdt+dB_t^{a,b}$.
  In this work, we provide least squares-type estimators for $\theta$ based
continuous-time and discrete-time observations of $X$. The strong consistency
and the asymptotic behavior in distribution of the estimators are studied for
all $(a,b)$ such that $a>-1$, $|b|<1$, $|b|<a+1$. Here we extend the results of
\cite{SYY2,SYY} (resp. \cite{CSC}), where the strong consistency and the
asymptotic distribution of the estimators are proved for
  $-\frac12<a<0$, $-a<b<a+1$ (resp. $-1<a<0$, $-a<b<a+1$).
