  We construct an infinite family of two-Lee-weight and three-Lee-weight codes
over the chain ring $\mathbb{F}_p+u\mathbb{F}_p.$ They have the algebraic
structure of abelian codes. Their Lee weight distribution is computed by using
Gauss sums. Then by using a linear Gray map, we obtain an infinite family of
abelian codes with few weights over $\mathbb{F}_p$. In particular, we obtain an
infinite family of two-weight codes which meets the Griesmer bound with
equality. Finally, an application to secret sharing schemes is given.
