We show that black holes grow substantially by accretion at close to Eddington rates. Using a complete sample of soft X-ray selected AGNs, Grupe & Mathur (2004, ApJ, 606, L41) have shown that narrow line Seyfert 1 galaxies, as a class, lie below the MBH-σ relation of normal galaxies. Some NLS1s, however, lie on or close to the MBH-σ relation. Here we show that not all NLS1s accrete at close to Eddington rates: those with low L/LEddington are close to the MBH-σ relation, and those with high L/LEddington are far from it. With various tests in this paper, we argue that black holes grow in mass substantially in their high-accretion phase and approach the MBH-σ relation over time. The mass growth in a low accretion phase, as in BLS1s and also in some NLS1s, appears to be insignificant. Any theoretical model attempting to explain the MBHσ relation needs to explain the above observations.